# Intermediate field-induced phase of the honeycomb magnet BaCo2(AsO4)2

Prashanta K. Mukharjee<EMAIL_ADDRESS>Bin Shen  Sebastian Erdmann
Anton Jesche  Julian Kaiser  Experimental Physics VI, Center for Electronic
Correlations and Magnetism, Institute of Physics, University of Augsburg,
86159 Augsburg, Germany  Priya R. Baral  Oksana Zaharko  Laboratory for
Neutron Scattering and Imaging (LNS), Paul Scherrer Institute (PSI), CH-5232,
Villigen, Switzerland  Philipp Gegenwart<EMAIL_ADDRESS>Experimental Physics VI, Center for Electronic Correlations and Magnetism,
Institute of Physics, University of Augsburg, 86159 Augsburg, Germany
Alexander A. Tsirlin<EMAIL_ADDRESS>Felix Bloch Institute for Solid-
State Physics, University of Leipzig, 04103 Leipzig, Germany

###### Abstract

We use magnetometry, calorimetry, and high-resolution capacitive dilatometry,
as well as single-crystal neutron diffraction to explore temperature-field
phase diagram of the anisotropic honeycomb magnet BaCo2(AsO${}_{4})_{2}$. Our
data reveal four distinct ordered states observed for in-plane magnetic
fields. Of particular interest is the narrow region between 0.51 and 0.55 T
that separates the up-up-down order from the fully polarized state and
coincides with the field range where signatures of the spin-liquid behavior
have been reported. We show that magnetic Bragg peaks persist in this
intermediate phase, thus ruling out its spin-liquid nature. However, the
simultaneous nonmonotonic evolution of nuclear Bragg peaks suggests the
involvement of the lattice, witnessed also in other regions of the phase
diagram where large changes in the sample length are observed upon entering
the magnetically ordered states. Our data highlight the importance of lattice
effects in BaCo2(AsO${}_{4})_{2}$.

Introduction. Quantum spin liquid (QSL) is an exotic state of matter in which
strong quantum fluctuations prevent magnetic long-range order (LRO) down to
very low temperatures [1]. The exactly solvable $S=\frac{1}{2}$ honeycomb
Kitaev model possessing bond-dependent nearest-neighbor Ising-type
interactions offers a promising venue for stabilizing the QSL ground state
with the excitation spectrum described by emergent Majorana fermions and gauge
fluxes [2]. Experimental realization of this model has been of significant
interest. The $d^{5}$ (Ir4+, Ru3+) [3] and, more recently, $d^{7}$ (Co2+) [4]
transition-metal ions were proposed as suitable building blocks of the Kitaev
magnets.

The majority of Ir-, Ru-, and Co-based honeycomb materials show long-range
order in zero magnetic field [5, 6]. However, it was conjectured that they may
lie in the vicinity of the QSL phase, and external field can be used to
suppress long-range order, thus giving way to the spin liquid. Indeed,
magnetic field applied along a suitable direction leads to a rapid suppression
of the ordered phase in $\alpha$-RuCl3 and $\beta$-Li2IrO3 above the critical
fields of $B_{c}\simeq 7$ T and 2.8 T, respectively [7, 8, 9]. The behavior of
$\alpha$-RuCl3 above $B_{c}$ remains vividly debated. Whereas quantized
thermal Hall effect [10, 11, 12], oscillations in thermal conductivity [13,
14, 15], and a distinct excitation continuum [16, 17] observed in this field
range might be vestiges of the spin-liquid behavior, no distinct phase
associated with this putative spin liquid could be identified [18, 19, 20,
21].

Here, we focus on the Co-based honeycomb magnet, BaCo2(AsO${}_{4})_{2}$
(BCAO), that was recently proposed as a possible Kitaev candidate [22], and
address its behavior right above $B_{c}$ in the same region where putative
signatures of the spin-liquid physics of $\alpha$-RuCl3 have been revealed.
The two materials show different sequences of magnetically ordered states,
double-zigzag-like and field-induced up-up-down (uud) orders in BCAO [23, 24,
25] vs. different types of zigzag order in $\alpha$-RuCl3 [26]. Moreover, a
detailed study of spin dynamics [27], along with ab initio calculations [28],
suggested that an easy-plane $J_{1}-J_{3}$ Hamiltonian is more suitable for
BCAO than an extended Kitaev model commonly accepted for $\alpha$-RuCl3 [5].
On the other hand, both materials show several apparent similarities. They can
be polarized by moderate in-plane fields and demonstrate an unusual behavior
right above $B_{c}$. The critical field of BCAO is as low as $B_{c}\simeq 0.5$
T. An excitation continuum [29] and a small linear contribution to the thermal
conductivity [30], which is often associated with spinon excitations of the
spin liquid, were recently reported in this field range, suggesting that BCAO
might also be tuned toward a spin-liquid phase by the applied field.

In the following, we present the thermodynamic and microscopic
characterization of BCAO with the special focus on the field range around
$B_{c}$. Using several thermodynamic probes, we map out the temperature-field
phase diagram of this material and uncover the previously overlooked
intermediate phase that appears near $B_{c}$ and separates the uud order from
the fully polarized state. This intermediate phase coincides with the field
range where both excitation continuum and linear-in-temperature thermal
conductivity have been reported. However, it does not show characteristic
signatures of a spin liquid, namely, the absence of the long-range magnetic
order because the magnetic Bragg peak persists in this phase. The formation of
such an intermediate phase is not compatible with the existing theories of
BCAO. Its stabilization may involve lattice effects, which are also prominent
in other regions of the phase diagram where our data evidence large changes in
the sample length across several phase transitions.

Figure 1: Magnetic field-temperature $(HT)$ phase diagram of BaCo2(AsO4)2 for
$B\perp c$ constructed from magnetization $M(T,B)$, dilatometry $L(T,B)$ and
MCE measurements. Phase diagrams for different in-plane field directions are
shown in Fig. S11. Inset: A honeycomb motif containing CoO6 octahedra and AsO4
tetrahedra at the center. The individual Co-honeycomb layers are stacked along
the $c$-axis with Ba atoms separating them.

Methods. Dark-pink-colored single crystals of BCAO were synthesized by the
flux method [31]. High quality of these crystals was confirmed by an extensive
mapping with Laue diffraction and by neutron diffraction (see Fig. S1).
Temperature- and field-dependent magnetization was measured using the
commercial SQUID magnetometer (MPMS3, Quantum Design), with orientation-
dependent magnetization data collected using the sample rotation stage of
MPMS3. Specific heat was measured using a standard relaxation technique in
Quantum Design Physical Properties Measurement System (QD - PPMS). The
magnetic Grüneisen parameter ($\Gamma_{m}$), which equals the adiabatic
magnetocaloric effect (MCE), was determined by the alternating-field method
[32]. Thermal expansion was measured with the aid of a compact ultrahigh-
resolution capacitive dilatometer in the QD PPMS [33]. The length change as a
function of temperature or field was recorded using a capacitance bridge
(Andeen Haggerling 2550A). Neutron diffraction data from a single crystal were
collected on the ZEBRA diffractometer at SINQ, Paul Scherrer Institute.

Phase diagram. Previous report revealed an almost isotropic behavior of BCAO
within the honeycomb plane [27], in contrast to $\alpha$-RuCl3 that showed
different phase boundaries for two nonequivalent in-plane field directions,
$b$ and $b^{*}$ [21, 26]. Our data for BCAO confirm this isotropic behavior.
Therefore, in Fig. 1 we merged the data points obtained for different field
directions, whereas individual phase diagrams for $B\|b$ and $B\|b^{*}$ can be
found in the Supplemental Material [Fig. S11]. In magnetometry, we determine
temperature and field values from the peak positions of their respective
derivatives. In dilatometry, we consider the peak positions of linear thermal
expansion coefficient ($\alpha$) and magnetostriction ($\lambda$), while in
the MCE ($\Gamma_{m}$) data, we identify the transitions from the positions of
the minima and zero-crossings [21].

Two magnetically ordered states of BCAO, the incommensurate double-zigzag
order and commensurate uud order [23, 24, 25], are labeled as phases I and III
in our phase diagram, respectively. Phase V is the fully polarized state.
Additionally, we uncover phases II and IV that separate I from III and III
from V, respectively. The main evidence for phase II comes from ac-
susceptibility measurements, whereas phase IV is most crisply seen from
dilatometry. This phase IV is of particular interest, as it at least partially
coincides with the field range where signatures of the spin-liquid physics
have been reported [29, 30].

Figure 2: (a) Magnetic field dependence of $\chi_{ac}^{\prime}$ at 2 K for
$B||b$. The arrow marks indicate position of respective phase boundaries. (b)
Angle-dependent magnetic susceptibility measured at $T=2.4$ K, for different
magnetic fields while rotating the crystal in the $ab$ plane.

Temperature-dependent dc-magnetization of BCAO [Fig. S2] shows an abrupt drop
at $T_{N}$ followed by a broad maximum that appears between 0.12 and 0.26 T,
the field range that we identify as phase II. The boundaries of this phase are
most clearly visible in the ac-susceptibility that shows two consecutive peaks
as a function of field (Fig. 2a). The lower boundary of phase II is
characterized by the abrupt increase in $M(B)$, suggesting that the
transformation from the antiferromagnetic double-zigzag into the partially
polarized uud state starts upon entering phase II. This transformation is
first-order in nature, as witnessed by the large field hysteresis [Fig. S4].
The anomalies associated with phase II disappear around 4 K, whereas phases I
and III survive up to $5.2-5.3$ K.

The transition between I and III involves a flip of ferromagnetic zigazag spin
chains from the
$\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\downarrow$
into the $\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow$ configuration,
which is essentially the creation and shift of domain walls. It is plausible
that such a shift requires thermal energy and at lower temperatures involves
an intermediate region, which we identify as phase II. This phenomenology may
be similar to the isostructural BaCo2(PO${}_{4})_{2}$ where magnetization
curve shows distinct steps that correspond to discrete shifts of the domain
walls, albeit at temperatures below 1 K only [34].

Figure 3:  Thermal expansion and magnetostriction plots of BCAO. (a)
Normalized relative length changes vs temperature measured while warming at
various fields. The curves are shifted vertically for clarity. (b) Plot of the
linear TE coefficient ($\alpha$) in different fields. For $B=0$ T, in phase I,
$\alpha$ shows a positive peak. As we move into phase II, $\alpha$ bifurcates
to two peaks with opposite sign as indicated by the arrows. (c) Normalized
relative length changes vs magnetic field measured while sweeping magnetic
field up at several temperatures above and below $T_{N}$. The light grey line
is a guide to eye showing the systematic shift of minimum around $B_{c}$. (d)
Plot of linear MS coefficient $\lambda$ vs. $B$ at $T=2$ K. Two prominent
peaks are indicated by arrows around $B_{c}$. The transition around 0.15 T
exhibits significant hysteresis, consistent with the field-dependent
differential susceptibility [inset], reflecting its first-order nature.

We now turn to thermal expansion (TE) and magnetostriction (MS) where all
phase transitions are clearly visible thanks to the large lattice effects
involved. BCAO shrinks along $c$ upon entering phase I and expands along $c$
upon entering phase III, but the most interesting behavior is seen in
temperature-dependent thermal expansion measured above 0.2 T where sample
length changes non-monotonically indicating two consecutive phase transitions,
first with the decrease and then with the increase in $c$ upon cooling (Fig.
3a). A similar behavior is seen in field-dependent sample length that changes
non-monotonically on going from phase III into phase V (Fig. 3c). The two
transitions can be tracked using the minima and maxima in linear thermal
expansion coefficient $\alpha=(1/L_{0})(d\Delta L/dT)$ and magnetostriction
$\lambda=(1/L_{0})(d\Delta L/dB)$ (Fig. 3b,d), resulting in the distinct
region of phase IV that envelops phase III and separates it from phase V. This
phase IV is remarkably different from phase II because it extends in
temperature all the way up to 5.3 K where BCAO enters its paramagnetic state.
Moreover, the transitions associated with phase IV should be second-order or
weakly first-order in nature, as they do not show any significant hysteresis.

Intermediate phase. Phase IV is most clearly seen in the thermal expansion and
magnetostriction data, whereas field-dependent dc-magnetization and ac-
susceptibility merely show a broad peak around $B_{c}$, which is consistent
with the presence of two consecutive transitions but does not prove the
existence of an intermediate phase. This problem is at least partially
mitigated in the MCE measurement where $\Gamma_{m}$ displays the minimum and
zero-crossing at the III–IV and IV–V boundaries, respectively [see Fig. S9].
It independently proves the existence of the intermediate phase. Additionally,
we note that the symmetry of BCAO should change between phases III and IV.
Fig. 2b shows angle-dependent susceptibility measured at different magnetic
fields. While phase I is characterized by the quite low symmetry, phase III
demonstrates the 6-fold symmetry expected for the uud state in the $R\bar{3}$
crystal structure. At 0.5 T, on approaching phase IV, the hexagon is turned by
$30^{\circ}$, such that the minima and maxima of the susceptibility are
swapped. It is also deformed [see Fig. S5b], indicating the loss of the 6-fold
symmetry.

Figure 4:  Magnetic field dependence of magnetic and nuclear Bragg peak at
$\mathbf{q}=(\frac{1}{3},0,-\frac{4}{3})$, and (3,0,0) in phase IV for
$B\parallel b$.

We further probed phase IV using neutron diffraction. In phase III, we
observed the magnetic Bragg peak at $\mathbf{q}=(\frac{1}{3},0,-\frac{4}{3})$
in agreement with the previous studies [24, 27]. This peak persists in phase
IV but gradually loses its intensity and vanishes when the fully polarized
state is reached [see Fig. 4]. At first glance, this behavior may suggest that
phase IV is simply a two-phase region where the uud and fully polarized states
coexist. In this case, one expects the gradual increase in the intensity of
$\mathbf{q}=0$ peaks through the shift of the magnetic intensity from
$\mathbf{q}=(\frac{1}{3},0,-\frac{4}{3})$ to $\mathbf{q}=0$ as magnetization
increases. Some of the $\mathbf{q}=0$ peaks indeed show abrupt changes around
0.51 and 0.55 T, but their intensity within phase IV changes non-monotonically
as shown in Fig. 4. A lattice deformation may thus accompany the formation of
phase IV, in agreement with the loss of the 6-fold symmetry in the angle-
dependent susceptibility (see Fig. 2b). Field-induced intermediate phases
stabilized by the spin-lattice coupling have been indeed reported in
pyrochlore [35, 36] and triangular [37] magnets, albeit at much higher
magnetic fields.

Pressure dependence of the transitions. BCAO shows remarkably large lattice
effects upon its magnetic transitions. The zero-field transition toward phase
I is accompanied by the relative length change of $4\times 10^{-4}$, about
four times larger than in $\alpha$-RuCl3 [20]. Despite such a strong coupling
to the lattice, this transition appears as purely second-order, as it does not
show any thermal hysteresis [Fig. S7]. The large lattice effects suggest that
different phases of BCAO may be sensitive to pressure.

The peaks in $\alpha$ and $\lambda$ yield the initial pressure dependence of
the transitions. According to Maxwell’s relation, $\lambda V=-(dM/dp)_{T,B}$
where $V$ is the volume and $M$ is the magnetization. Whereas we do not
measure volume magnetostriction in our experiments, the qualitative behavior
given by the positive ($\lambda_{c}>0$) and negative peaks ($\lambda_{c}<0$)
in linear magnetostriction suggests that magnetization should decrease upon
compression in phase I, whereas it should increase in phase III. The former
trend is indeed confirmed by our isothermal magnetization measurements
performed under hydrostatic pressure [Fig. S10c].

Pressure dependence of the transition temperature, $dT_{N}/dp$, is given by
the Clausius-Clapeyron and Ehrenfest relations for the first- and second-order
transitions, respectively. From the maxima and minima in $\alpha_{c}$, we
expect $T_{N}$ to increase with pressure in phase I and decrease with pressure
in phase III. These trends are again consistent with the direct magnetization
measurements under hydrostatic pressure using both our data [31] and the
earlier report [38]. Using the discontinuities of the volume thermal expansion
coefficient and the specific heat measured in zero field, we obtain from the
Ehrenfest relation $(dT_{N}/dp)_{p\rightarrow 0}\simeq 0.86$ K/GPa, which is
several times larger than 0.3 K/GPa from our direct measurement of $T_{N}$
under hydrostatic pressure [31]. This discrepancy probably indicates that the
full pressure dependence of $T_{N}$ up to 2 GPa deviates from the initial
pressure dependence given by the ambient-pressure specific heat and thermal
expansion. Indeed, the phase diagram of BCAO may drastically change under
pressure because the $\frac{1}{3}$-plateau indicative of the uud order is no
longer seen at 1.6 GPa [31]. It means that phase III gives way to some other
phase that allows a gradual increase in the magnetization, similar to phase IV
at ambient pressure. Whether phase IV becomes the dominant field-induced state
of BCAO under pressure would be an interesting question for further
investigation.

Discussion. Our data reveal the previously overlooked aspect of BCAO, its
narrow intermediate phase around $B_{c}$. A somewhat similar phase has been
reported in $\alpha$-RuCl3 [26, 21] where maxima and minima of the angle-
dependent susceptibility are also swapped upon entering this intermediate
phase, around 6 T [26]. The intermediate phase of $\alpha$-RuCl3 has been
understood as the change from the three-layer periodicity of zigzag order in
low fields toward the six-layer periodicity as the result of anisotropic
couplings between the layers [26]. Examining a similar scenario for BCAO is
hindered by the broadness of its magnetic Bragg peaks along $l$ [27],
presumably due to the frustrated interlayer order. However, within the
resolution of our measurement we observed neither the shift nor a change in
the shape of the magnetic Bragg peak, which remains centered around
$l\simeq-\frac{4}{3}$ across phase IV. The change in the interlayer order and
the simple repetition of the $\alpha$-RuCl3 scenario in BCAO are thus
unlikely.

Theoretical studies suggest that BCAO can be described by an easy-plane
$J_{1}-J_{3}$ model that hosts the double-zigzag ground state in zero field as
a narrow region between the ferromagnetic and zigzag states [39, 40, 41].
Assuming that BCAO falls into this narrow region of the double-zigzag order,
application of the magnetic field should produce the uud state followed by the
fully polarized state, either directly or via a phase with canted V-type order
[42]. However, this V-type order should be characterized by a different
position of the magnetic Bragg peak in the $hk$ plane compared to the uud
state. Therefore, our data also rule out V-type order as the possible
intermediate phase in this case.

The intermediate phase observed in our work does not cover the full field
range where indications of the spin-liquid physics have been reported. Indeed,
at least the linear-in-temperature contribution to the thermal conductivity
extends up to 0.6 T [30], well into the fully polarized state. However, the
onset of this unusual behavior at 0.50 T is clearly correlated with the
formation of phase IV and not with reaching the fully polarized state around
0.55 T, so the presence of this intermediate phase must be taken into account
when the microscopic scenario of the spin-liquid-like behavior of BCAO is
looked for.

Conclusions. Our comprehensive study of the anisotropic honeycomb magnet
BaCo2(AsO4)2 reveals an intermediate phase forming in this material near
$B_{c}$, in the same field range where signatures of the spin-liquid behavior
have been reported. While at first glance this behavior may look similar to
the renowned $\alpha$-RuCl3, our data suggest a different microscopic
scenario. We show that signatures of the spin-liquid behavior observed in the
field range near $B_{c}$ should not be associated with the formation of a
separate spin-liquid phase, because magnetic Bragg peaks persist throughout
phase IV. Substantial lattice effects accompany the formation of this phase
and should be important for understanding the intriguing physics of BCAO.

###### Acknowledgements.

We thank Pavel Maksimov, Sasha Chernyshev, Lukas Janssen, and Radu Coldea for
fruitful discussions. P.K.M was supported by the Alexander von Humboldt
foundation. The neutron single cyrstal diffraction experiment was performed at
SINQ, Paul Scherrer Institute, Villigen, Switzerland. This work was funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – TRR
360 – 492547816.

## References

  * Broholm  _et al._ [2020] C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quantum spin liquids, Science 367, eaay0668 (2020).
  * Kitaev [2006] A. Kitaev, Anyons in an exactly solved model and beyond, Annals of Physics 321, 2 (2006).
  * Jackeli and Khaliullin [2009] G. Jackeli and G. Khaliullin, Mott insulators in the strong spin-orbit coupling limit: From Heisenberg to a quantum compass and Kitaev models, Phys. Rev. Lett. 102, 017205 (2009).
  * Liu and Khaliullin [2018] H. Liu and G. Khaliullin, Pseudospin exchange interactions in ${d}^{7}$ cobalt compounds: Possible realization of the Kitaev model, Phys. Rev. B 97, 014407 (2018).
  * Winter  _et al._ [2017] S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valentí, Models and materials for generalized Kitaev magnetism, J. Phys.: Condens. Matter 29, 493002 (2017).
  * Kim  _et al._ [2022] C. Kim, H.-S. Kim, and J.-G. Park, Spin-orbital entangled state and realization of kitaev physics in $3d$ cobalt compounds: a progress report, J. Phys.: Condens. Matter 34, 023001 (2022).
  * Takagi  _et al._ [2019] H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and realization of Kitaev quantum spin liquids, Nature Reviews Physics 1, 264 (2019).
  * Ruiz  _et al._ [2017] A. Ruiz, A. Frano, N. P. Breznay, I. Kimchi, T. Helm, I. Oswald, J. Y. Chan, R. J. Birgeneau, Z. Islam, and J. G. Analytis, Correlated states in $\beta$-Li2IrO3 driven by applied magnetic fields, Nature Comm. 8, 961 (2017).
  * Majumder  _et al._ [2019] M. Majumder, F. Freund, T. Dey, M. Prinz-Zwick, N. Büttgen, Y. Skourski, A. Jesche, A. Tsirlin, and P. Gegenwart, Anisotropic temperature-field phase diagram of single crystalline $\beta$-Li2IrO3: Magnetization, specific heat, and 7Li NMR study, Phys. Rev. Materials 3, 074408 (2019).
  * Yokoi  _et al._ [2021] T. Yokoi, S. Ma, Y. Kasahara, S. Kasahara, T. Shibauchi, N. Kurita, H. Tanaka, J. Nasu, Y. Motome, C. Hickey, S. Trebst, and Y. Matsuda, Half-integer quantized anomalous thermal Hall effect in the Kitaev material candidate $\alpha$-RuCl3, Science 373, 568 (2021).
  * Lefrançois  _et al._ [2022] E. Lefrançois, G. Grissonnanche, J. Baglo, P. Lampen-Kelley, J.-Q. Yan, C. Balz, D. Mandrus, S. E. Nagler, S. Kim, Y.-J. Kim, N. Doiron-Leyraud, and L. Taillefer, Evidence of a phonon Hall effect in the Kitaev spin liquid candidate $\alpha$-RuCl3, Phys. Rev. X 12, 021025 (2022).
  * Bruin  _et al._ [2022a] J. A. N. Bruin, R. R. Claus, Y. Matsumoto, N. Kurita, H. Tanaka, and H. Takagi, Robustness of the thermal Hall effect close to half-quantization in $\alpha$-RuCl3, Nature Phys. 18, 401 (2022a).
  * Czajka  _et al._ [2021] P. Czajka, T. Gao, M. Hirschberger, P. Lampen-Kelley, A. Banerjee, J. Yan, D. G. Mandrus, S. E. Nagler, and N. P. Ong, Oscillations of the thermal conductivity in the spin-liquid state of $\alpha$-RuCl3, Nature Phys. 17, 915 (2021).
  * Lefrançois  _et al._ [2023] E. Lefrançois, J. Baglo, Q. Barthélemy, S. Kim, Y.-J. Kim, and L. Taillefer, Oscillations in the magnetothermal conductivity of $\alpha$-RuCl3: Evidence of transition anomalies, Phys. Rev. B 107, 064408 (2023).
  * Bruin  _et al._ [2022b] J. A. N. Bruin, R. R. Claus, Y. Matsumoto, J. Nuss, S. Laha, B. V. Lotsch, N. Kurita, H. Tanaka, and H. Takagi, Origin of oscillatory structures in the magnetothermal conductivity of the putative Kitaev magnet $\alpha$-RuCl3, APL Materials 10, 090703 (2022b).
  * Banerjee  _et al._ [2018] A. Banerjee, P. Lampen-Kelley, J. Knolle, C. Balz, A. A. Aczel, B. Winn, Y. Liu, D. Pajerowski, J. Yan, C. A. Bridges, A. T. Savici, B. C. Chakoumakos, M. D. Lumsden, D. A. Tennant, R. Moessner, D. G. Mandrus, and S. E. Nagler, Excitations in the field-induced quantum spin liquid state of $\alpha$-RuCl3, npj Quantum Materials 3, 8 (2018).
  * [17] A. Sahasrabudhe, M. A. Prosnikov, T. C. Koethe, P. Stein, V. Tsurkan, A. Loidl, M. Grüninger, H. Hedayat, and P. H. M. van Loosdrecht, Evidence for gapless quantum spin liquid in a honeycomb lattice, arXiv:2305.03400 .
  * Bachus  _et al._ [2020] S. Bachus, D. A. S. Kaib, Y. Tokiwa, A. Jesche, V. Tsurkan, A. Loidl, S. M. Winter, A. A. Tsirlin, R. Valentí, and P. Gegenwart, Thermodynamic perspective on field-induced behavior of $\alpha$-RuCl3, Phys. Rev. Lett. 125, 097203 (2020).
  * Schönemann  _et al._ [2020] R. Schönemann, S. Imajo, F. Weickert, J. Yan, D. G. Mandrus, Y. Takano, E. L. Brosha, P. F. S. Rosa, S. E. Nagler, K. Kindo, and M. Jaime, Thermal and magnetoelastic properties of $\alpha$-RuCl3 in the field-induced low-temperature states, Phys. Rev. B 102, 214432 (2020).
  * Gass  _et al._ [2020] S. Gass, P. M. Cônsoli, V. Kocsis, L. T. Corredor, P. Lampen-Kelley, D. G. Mandrus, S. E. Nagler, L. Janssen, M. Vojta, B. Büchner, and A. U. B. Wolter, Field-induced transitions in the Kitaev material $\alpha$-RuCl3 probed by thermal expansion and magnetostriction, Phys. Rev. B 101, 245158 (2020).
  * Bachus  _et al._ [2021] S. Bachus, D. A. S. Kaib, A. Jesche, V. Tsurkan, A. Loidl, S. M. Winter, A. A. Tsirlin, R. Valentí, and P. Gegenwart, Angle-dependent thermodynamics of $\alpha$-RuCl3, Phys. Rev. B 103, 054440 (2021).
  * Zhong  _et al._ [2020] R. Zhong, T. Gao, N. P. Ong, and R. J. Cava, Weak-field induced nonmagnetic state in a Co-based honeycomb, Sci. Adv. 6, eaay6953 (2020).
  * Regnault  _et al._ [1977] L. Regnault, P. Burlet, and J. Rossat-Mignod, Magnetic ordering in a planar ${XY}$ model:BaCo2(AsO4)2, Physica B+C 86-88, 660 (1977).
  * Regnault and Rossat-Mignod [1979] L. P. Regnault and J. Rossat-Mignod, Effect of a magnetic field on the magnetic ordering of BaCo2(AsO${}_{4})_{2}$, J. Magn. Magn. Mater. 14, 194 (1979).
  * Regnault  _et al._ [2018] L.-P. Regnault, C. Boullier, and J. Lorenzo, Polarized-neutron investigation of magnetic ordering and spin dynamics in BaCo2(AsO4)2 frustrated honeycomb-lattice magnet, Heliyon 4, e00507 (2018).
  * Balz  _et al._ [2021] C. Balz, L. Janssen, P. Lampen-Kelley, A. Banerjee, Y. H. Liu, J.-Q. Yan, D. G. Mandrus, M. Vojta, and S. E. Nagler, Field-induced intermediate ordered phase and anisotropic interlayer interactions in $\alpha$-RuCl3, Phys. Rev. B 103, 174417 (2021).
  * Halloran  _et al._ [2023] T. Halloran, F. Desrochers, E. Z. Zhang, T. Chen, L. E. Chern, Z. Xu, B. Winn, M. Graves-Brook, M. B. Stone, A. I. Kolesnikov, Y. Qiu, R. Zhong, R. Cava, Y. B. Kim, and C. Broholm, Geometrical frustration versus Kitaev interactions in BaCo2(AsO${}_{4})_{2}$, Proc. Nat. Acad. Sci. 120, e2215509119 (2023).
  * Maksimov  _et al._ [2022] P. A. Maksimov, A. V. Ushakov, Z. V. Pchelkina, Y. Li, S. M. Winter, and S. V. Streltsov, Ab initio guided minimal model for the ”Kitaev” material BaCo2(AsO${}_{4})_{2}$: Importance of direct hopping, third-neighbor exchange, and quantum fluctuations, Phys. Rev. B 106, 165131 (2022).
  * Zhang  _et al._ [2023] X. Zhang, Y. Xu, T. Halloran, R. Zhong, C. Broholm, R. J. Cava, N. Drichko, and N. P. Armitage, A magnetic continuum in the cobalt-based honeycomb magnet BaCo2(AsO${}_{4})_{2}$, Nat. Mater. 22, 58 (2023).
  * [30] C. Tu, D. Dai, X. Zhang, C. Zhao, X. Jin, B. Gao, T. Chen, P. Dai, and S. Li, Evidence for gapless quantum spin liquid in a honeycomb lattice, arXiv:2212.07322 .
  * [31] See Supplemental Material for details of the experimental methods and sample characterization, as well as phase diagrams for different in-plane field directions.
  * Tokiwa and Gegenwart [2011] Y. Tokiwa and P. Gegenwart, High-resolution alternating-field technique to determine the magnetocaloric effect of metals down to very low temperatures, Rev. Sci. Instrum. 82, 013905 (2011).
  * Küchler  _et al._ [2017] R. Küchler, A. Wörl, P. Gegenwart, M. Berben, B. Bryant, and S. Wiedmann, The world’s smallest capacitive dilatometer, for high-resolution thermal expansion and magnetostriction in high magnetic fields, Rev. Sci. Instrum. 88, 083903 (2017).
  * Wang  _et al._ [2023] X. Wang, R. Sharma, P. Becker, L. Bohatý, and T. Lorenz, Single-crystal study of the honeycomb XXZ magnet BaCo2(PO${}_{4})_{2}$ in magnetic fields, Phys. Rev. Materials 7, 024402 (2023).
  * Miyata  _et al._ [2020] A. Miyata, H. Suwa, T. Nomura, L. Prodan, V. Felea, Y. Skourski, J. Deisenhofer, H.-A. K. von Nidda, O. Portugall, S. Zherlitsyn, V. Tsurkan, J. Wosnitza, and A. Loidl, Spin-lattice coupling in a ferrimagnetic spinel: Exotic $H-T$ phase diagram of MnCr2S4 up to 110 T, Phys. Rev. B 101, 054432 (2020).
  * Gen  _et al._ [2023] M. Gen, A. Ikeda, K. Aoyama, H. O. Jeschke, Y. Ishii, H. Ishikawa, T. Yajima, Y. Okamoto, X. Zhou, D. Nakamura, S. Takeyama, K. Kindo, Y. H. Matsuda, and Y. Kohama, Signatures of a magnetic superstructure phase induced by ultrahigh magnetic fields in a breathing pyrochlore antiferromagnet, Proc. Nat. Acad. Sci. 120, e2302756120 (2023).
  * Nakajima  _et al._ [2013] T. Nakajima, N. Terada, S. Mitsuda, and R. Bewley, Spin-driven bond order in a 1/5-magnetization plateau phase in the triangular lattice antiferromagnet CuFeO2, Phys. Rev. B 88, 134414 (2013).
  * Huyan  _et al._ [2022] S. Huyan, J. Schmidt, E. Gati, R. Zhong, R. J. Cava, P. C. Canfield, and S. L. Bud’ko, Hydrostatic pressure effect on the Co-based honeycomb magnet BaCo2(AsO${}_{4})_{2}$, Phys. Rev. B 105, 184431 (2022).
  * Jiang  _et al._ [2023] S. Jiang, S. R. White, and A. L. Chernyshev, Quantum phases in the honeycomb-lattice $J_{1}-J_{3}$ ferro-antiferromagnetic model, Phys. Rev. B 108, L180406 (2023).
  * Bose  _et al._ [2023] A. Bose, M. Routh, S. Voleti, S. K. Saha, M. Kumar, T. Saha-Dasgupta, and A. Paramekanti, Proximate Dirac spin liquid in the honeycomb lattice $J_{1}-J_{3}$ XXZ model: Numerical study and application to cobaltates, Phys. Rev. B 108, 174422 (2023).
  * [41] Y. Watanabe, S. Trebst, and C. Hickey, Frustrated ferromagnetism of honeycomb cobaltates: Incommensurate spirals, quantum disordered phases, and out-of-plane Ising order, arXiv:2212.14053 .
  * Maksimov [2023] P. A. Maksimov, Proximity-induced sequence of field transitions in the Kitaev candidate BaCo2(AsO${}_{4})_{2}$, Phys. Rev. B 108, L180405 (2023).
  * Yan  _et al._ [2019] J.-Q. Yan, S. Okamoto, Y. Wu, Q. Zheng, H. D. Zhou, H. B. Cao, and M. A. McGuire, Magnetic order in single crystals of Na3Co2SbO6 with a honeycomb arrangement of $3d^{7}$ Co2+ ions, Phys. Rev. Mater. 3, 074405 (2019).
  * Johnson  _et al._ [2015] R. D. Johnson, S. C. Williams, A. A. Haghighirad, J. Singleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O. Jeschke, R. Valentí, and R. Coldea, Monoclinic crystal structure of $\alpha$-RuCl3 and the zigzag antiferromagnetic ground state, Phys. Rev. B 92, 235119 (2015).
  * Rodríguez-Carvajal [1993] J. Rodríguez-Carvajal, Recent advances in magnetic structure determination by neutron powder diffraction, Physica B: Condens. Matter 192, 55 (1993).
  * Shen  _et al._ [2021] B. Shen, A. Jesche, M. L. Seidler, F. Freund, P. Gegenwart, and A. A. Tsirlin, Interplay of magnetism and dimerization in the pressurized kitaev material $\beta$-Li2IrO3, Phys. Rev. B 104, 134426 (2021).

Supplementary Material for
Intermediate field-induced phase of the honeycomb magnet BaCo2(AsO4)2

##  S-I Crystal Growth

Single crystals of BaCo2(AsO4)2 were grown by a similar method as reported in
[22]. Dark-pink, hexagonal plate-like crystals (inset of Fig. S1a) were
separated from the flux by washing in hot water. A few crystals were crushed
into powders, and room-temperature powder XRD data were collected. To
determine the sample orientation and possible stacking disorder, we collected
X-ray Laue back reflection patterns (Photonic Sciences) (Fig. S1b). The clear
Laue spots indicate the high quality of the BCAO single crystal. In contrast
to findings in other honeycomb materials [43], our Laue pattern clearly shows
the absence of rod-like diffuse spots, effectively ruling out the presence of
defects/stacking faults in BCAO. Moreover, through extensive Laue photography
covering the entirety of the crystal surface, we have confirmed that BCAO
crystals exhibit a monodomain structure, contrasting with the multi-domain
structure, which is often observed in $\alpha$-RuCl3 [44].

A few crystals were crushed into fine powder, and room-temperature x-ray
diffraction data were recorded using the Empyrean powder diffractometer from
PANalytical (CuKα radiation, $\lambda_{avg}\simeq 1.5418~{}\text{Å}$).
Rietveld refinement for the collected data performed using the ”FULLPROF”
software [45] confirms the single phase of BCAO and returns lattice constants
of $a=5.0052(2)~{}\text{Å}$ and $c=23.481(1)~{}\text{Å}$, which are consistent
with the previous report [22].

##  S-II Magnetization and Specific Heat

The evolution of $T_{N}$ was tracked from the magnetic susceptibility measured
in different applied magnetic fields for the $B\parallel b$ orientation as
shown in Fig. S2(a). In Fig. S2(b), we show a broad hump around $3.2$ K,
observed in the field range of 0.12 to 0.26 T and associated with phase II in
the phase diagram. The transition at $T_{N}\simeq 5.35$ K decreases
systematically before completely vanishing at $\sim 0.55$ T. Magnetic
isotherms collected for 5.5 K $\leq T\leq 2~{}$K are shown in Fig. S2(c). To
estimate the magnetic couplings between the Co2+ ions, we have performed
Curie-Weiss (CW) fit of the susceptibility data using
$\chi=\chi_{0}+C/(T-\theta)$, where $\chi_{0}$ is the temperature-independent
contribution, $C$ is the Curie constant, $\theta$ is the CW temperature. The
fit in the region $150-300$ K yields a positive CW temperature
$\theta_{\parallel,b}\simeq 37$ K suggesting predominant ferromagnetic
interactions. From the obtained value of $C$, the effective paramagnetic
moment ($\mu_{\rm eff})$ is estimated to be $\mu_{\rm eff\parallel,b}\simeq
5.58\mu_{B}$ [see Fig. S2(d)]. These estimations are in agreement with the
literature [22, 30]. The deviation of the obtained effective moment (taking
$=g\sqrt{J(J+1)}$, where $g$ is the Landé $g$-factor and $J=1/2$) from the
theoretical value for spin-$\frac{1}{2}$ ($1.73~{}\mu_{B}$) arises due to the
significant spin-orbit coupling, which is common in Co2+ quantum magnets.

The second-order nature of the transition into phase I is supported by
examining the $T$-dependent susceptibility (at $0.01$ T) throughout field-
cooled-cooling (FCC) and field-cooled-warming (FCW) processes [see Fig.
S2(e)]. The curves for both directions are reversible with no hysteresis. We
have also measured temperature-dependent specific heat for $B\parallel a^{*}$
as shown in Fig. S2(f). In zero-field, $C_{p}(T)$ shows a sharp peak at
$T_{N}$, which gradually decreases with increasing magnetic field and vanishes
above 0.5 T, in line with other experiments.

##  S-III Dilatometry

High-resolution capacitive dilatometry was used to measure length changes
$L(T,H)$. The experimental design is described in Ref. [33]. In all our
experiments, we measured the relative length changes along the $c$-direction,
while for field-dependent measurements, the magnetic field was oriented along
the $b$ and $b^{*}$-direction. The normalized relative length change is
defined as follows,

$\frac{\Delta
L(T,B)}{L_{0}}=\frac{L(T,B)-L(300~{}K,0~{}T)}{L(300~{}K,0~{}T)}.$ (S1)

Here, $L_{0}$ is the sample thickness at 300 K and 0 T. Throughout our
analysis, $\Delta L(T,B)$ values are standardized to the value at 2 K and 0 T.
For thermal expansion, the data were collected during cooling-warming cycles
with the sweep rate of 0.2 K/min. Magnetostriction measurements were performed
up to 1 T with the sweep rate of 60 mT/min. To calculate the pressure
dependence of $T_{N}$, we use the Ehrenfest relation for a second-order
transition,

$\frac{dT_{N}}{dp}=V_{m}T_{N}\frac{\Delta\beta}{\Delta C_{p}}$ (S2)

where $V_{m}$ = 2.54 $\times 10^{-5}$ m3/mol is the molar volume,
$\Delta\beta=\Delta\alpha_{a^{*}}+\Delta\alpha_{b}+\Delta\alpha_{c}$ is the
jump in the volume thermal expansion coefficient at $T_{N}$, and $\Delta
C_{p}$ is the jump in $C_{p}$ at $T_{N}$. For this purpose a crystal was cut
into a rectangular shape and corresponding thermal expansion and specific heat
are measured as shown in Fig. S6.

Figure S8(a) depicts the magnetostriction and field-dependent differential
susceptibility ($dM/dB$) together for $T=2$ K. The first-order transformation
(phase I $\leftrightarrow$ phase II) is evidenced by a sharp peak both in
$\lambda_{c}$ and $dM/dB$ plots. Near $B_{c}$ (in phase IV), $\lambda_{c}$
shows two peaks, whereas $dM/dB$ features a broad behavior. Figure S8(b) shows
the variation of $\lambda_{c}$ with magnetic fields for 5.5 K $\leq T\leq 2$
K.

##  S-IV Magnetocaloric Effect

Field dependence of the magnetic Grüneisen parameter ($\Gamma_{m}$) is used to
track field-induced states around $B_{c}$. This is done via the magnetocaloric
effect (MCE) measurement under quasiadiabatic conditions. The magnetic
Grüneisen parameter is defined as

$\Gamma_{m}=\frac{1}{T}\frac{\partial T}{\partial B}.$ (S3)

This method utilizes the alternating-field technique [32], wherein a weakly
oscillating magnetic field with an amplitude ($\Delta B$) is applied. This
induces oscillations in the sample temperature ($\Delta T$) due to the MCE. In
this method, the heat-capacity setup is directly employed, enabling
simultaneous measurement of both heat capacity and the Grüneissen parameter.
The variations of $\Gamma_{m}$ vs. $B$ for three temperatures are shown in
Fig. S9.

##  S-V Pressure-dependent Magnetization

Magnetization studies under pressure were performed in the QD MPMS SQUID
magnetometer. A single crystal of BCAO was loaded into the CuBe cell. Pressure
was determined by measuring the superconducting transition of a small piece of
Pb. Daphne oil was used as a pressure transmitting medium. The details of the
experimental procedure can be found in [46].

##  S-VI Single Crystal Neutron Diffraction

Single-crystal neutron diffraction was performed on the neutron diffractometer
ZEBRA at SINQ, PSI. The crystal of the size $3\times 3\times 0.3$ mm3 was
mounted into the 10 T vertical magnet with the $b$-axis vertical. Neutrons of
the wavelength 1.383 Å selected by the Ge-monochromator were used in the
normal beam geometry.

Figure S1: (a) Powder x-ray diffraction data for BaCo2(AsO4)2 collected at
$T=300$ K (black circles). The red lines indicate the Rietveld refinement fit
of the data. The Bragg peaks are denoted by green vertical bars and the bottom
blue line denotes the difference between the experimental and calculated
intensities. Inset: Dark pink crystals on a millimeter paper. (b) Laue pattern
of a crystal taken from the $c$ \- direction. The red spots represent
simulated Laue patterns generated using Crystal Maker software. (c) A
Schematic sketch of the convention used for different crystallographic
orientations. (d) Laue back-scattering grid analysis was utilized to map out
possible domains in the single crystal of BCAO. The diffraction images were
color-coded: orange for good diffraction images, green for weak patterns
(edges), and blue for no observable patterns (sample holder). It was confirmed
that different spots on the crystal surface had the same crystallographic
orientation, thus excluding multi-domain growth.  Figure S2: (a) Magnetic
susceptibility vs. temperature under magnetic fields up to 0.7 T. (b)
$\chi(T)$ measured for 0.12 T $\leq B\leq$ 0.26 T where a broad hump has been
observed. The green circles indicate the corresponding broad maximum
positions. (c) Magnetic isotherms measured for 5.5 K $\leq T\leq 2$ K. (d)
Inverse susceptibility vs. temperature along with the CW fit (red solid line).
(d) Inverse susceptibility vs. temperature along with CW fit (red solid line).
(e) $\chi(T)$ measured at $B=0.01~{}$T for FCC-FCW cycle. (f) Specific-heat
divided by temperature [$C_{p}/T$] vs. $T$ for 0 T $\leq H\leq$ 0.7 T.  Figure
S3: Magnetic field dependence of $\chi_{ac}^{\prime}$ for 5.5 K $\leq T\leq 2$
K for $B\parallel b$. Three clear transitions marked as $B_{1}$, $B_{2}$, and
$B_{3}$ are observed at 2 K. As temperature increases up to $T_{N}$: $B_{1}$
initially remains constant, then steadily increases, $B_{2}$ remains stable
until 4 K, then abruptly disappears above this, $B_{3}$ exhibits a consistent
decrease beyond 3 K.  Figure S4: (a,b) Magnetic field dependence of
differential susceptibility ($dM/dB$) and $\chi_{ac}^{\prime}$ for both up and
down sweep cycles at $2$ K. The arrow marks indicates positions of $B_{1}$,
$B_{2}$, and $B_{3}$ similar to Fig. S3. (c,d) Enlarged plots of
$\chi_{ac}^{\prime}$ and $dM/dB$ in phase IV regime where a broad nature has
been observed.  Figure S5: (a) Definition of the rotation angle $\phi$ used
for the angle-dependent susceptibility. (b) Linear plots of angle-dependent
magnetic susceptibility at different fields. The clear exchange between the
minima and maxima is seen on going between phases III and IV.  Figure S6:
(a,b,c) Plots of the linear TE coefficient $\alpha$ vs. $T$ for the three
crystallographic directions. (b) Zero-field specific heat vs. $T$ for 2 K
$\leq T\leq$ 10 K.  Figure S7: Relative length change measured at zero
magnetic field on cooling and warming shows reversibility and indicates
second-order nature of the phase transition in zero field. No anomalies were
observed above 10 K up to 300 K.  Figure S8: (a) Magnetic field dependence of
$\lambda_{c}$ and $dM/dB$ plotted in left and right $y$-axes respectively at
$T=2$ K for the sweep up field cycle. (b) Plots of $\lambda_{c}$ vs. $B$
measured up to $1$ T for 5.5 K $\leq T\leq 2$ K.  Figure S9: Magnetic field
dependence of the magnetic Grüneissen parameter in phase IV. The two
transitions associated with phase IV can be identified from the peak and zero
crossing of the $\Gamma_{m}$ plots at each temperatures.  Figure S10: (a,b)
Temperature-dependent DC magnetic susceptibility $\chi(T)$ measured under
different pressures from 2 to 10 K under 0.1 and 0.3 T fields. Inset:
Corresponding pressure–temperature phase diagram. (c) Magnetic isotherms
measured up to 1 T at 2 K for different applied pressures.  Figure S11:  (a,b)
Field-temperature phase diagrams of BCAO constructed from the magnetometry and
dilatometry measurements, respectively. (c) Schematic presentation of phases
I, III, and V in BCAO.


# The fork and its role in unification
of closure algebras

Ivo Düntsch
Dept. of Computer Science,
Brock University, St Catharines, Canada,
<EMAIL_ADDRESS>Wojciech Dzik
Institute of Mathematics,
University of Silesia, Katowice, Poland,
<EMAIL_ADDRESS>
###### Abstract

We consider the two-pronged fork frame $F$ and the variety
$\operatorname{\mathbf{Eq}}(B_{F})$ generated by its dual closure algebra
$B_{F}$. We describe the finite projective algebras in
$\operatorname{\mathbf{Eq}}(B_{F})$ and give a purely semantic proof that
unification in $\operatorname{\mathbf{Eq}}(B_{F})$ is finitary and not
unitary.

##  1 Introduction

Unification of (first order) terms was introduced by J.A.Robinson (1976) as
the basic operation of the resolution principle used in automated theorem
provers. Nowadays unification plays an essential role in many applications of
logic to Computer Science, especially in Automated Deduction. Unification
theory for equational theories and logics is an important topic, for example,
for automatic deduction and rewriting systems. It has been a well researched
concept for some time; for details we refer the reader to the chapter by
Baader and Snyder, [2] in the Handbook of Automated Reasoning, and for an
early account of the role of unification in Computer Science we point the
reader to the overview by Stan Burris, [7]. Algebraic unification via
projective algebras was introduced and investigated by Ghilardi, [16]. He
showed that for any equational theory the symbolic and the algebraic
unification types coincide via a suitable translation. He concludes that,
broadly speaking, unification in varieties depends only on finitely presented
projective algebras. The syntactic approach to unification often requires long
and complicated calculations, while the algebraic approach via projective
algebras often offers much simpler solutions. It is particularly useful for
the determination of the unification type of a given theory or a logic.
Another important step (or aspect) consists of effectiveness of this
determination.

In this paper the algebraic approach via projective algebras is demonstrated
to determine the unification type of varieties of closure algebras. In
particular, we investigate in detail the equational theory of the complex
algebra $B_{F}$ of the 2-pronged fork $F$, shown in Figures 2 and 2. Elements
closed in $B_{F}$ are shown as bullets $\bullet$.

Figure 1: The 2-pronged fork $F$
$\textstyle{v}$$\textstyle{w}$$\textstyle{u\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Figure 2: The complex algebra $B_{F}$
$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet}$

The 2-pronged fork, or simply, fork, and its logic $L(F)$ have received some
interest in the literature: Aiello et al., [1] have provided an axiomatization
of $\operatorname{\mathbf{Eq}}(B_{F})$, Dzik, [10] proved that it does not
have unitary unification, and recently Dzik et al., [11] showed that the logic
has finitary unification. The unification results were proved by $n$-Kripke
models and syntactic means, and in this paper we provide purely semantic
proofs using the duality between finite closure algebras and finite
quasiorders; along the way we provide a characterization of the finite
projective algebras in $\operatorname{\mathbf{Eq}}(B_{F})$. We also give a
concrete example of an algebra in $\operatorname{\mathbf{Eq}}(B_{F})$ which
has finitary and not unitary unification. As the algebraic approach may not be
familiar to all readers, we shall briefly outline its concepts tailored to our
situation in Section 3.

##  2 Notation and first definitions

The identity function on a set is denoted by $id$, and the identity relation
by $1^{\prime}$. A _frame_ is a structure $\langle W,R\rangle$ where $W$ is a
non-empty set and $R$ a binary relation on $W$; we denote the converse of $R$
by $R^{\smile{}~{}}$. If $x\in W$ we set $R(x):=\\{y\in W:x\mathrel{R}y\\}$. A
non-empty subset $U$ of $W$ is _connected_ (in the sense of graph theory) if
for all $x,y\in U$ there is an $R\cup R^{\smile{}~{}}$ path from $x$ to $y$. A
maximally connected subset of $W$ is called a _connected component_ , or just
_component_. $R$ is _rooted_ if there is some $x\in W$ such that there is an
$R$-path from $x$ to any element of $W$.

A _generated subframe_ of $\langle W,R\rangle$ is a structure $\langle
V,S\rangle$ such that $\langle V,S\rangle$ is a first order substructure of
$\langle W,R\rangle$ and satisfies

(2.1) $\displaystyle(\forall u,v)[u\in W\text{ and }u\mathrel{R}v\
\mathrel{\Rightarrow}v\in V].$

If $\langle V,S\rangle$ is (isomorphic to) a generated substructure of
$\langle W,R\rangle$ we write $\langle
V,S\rangle\overset{g}{\hookrightarrow}\langle W,R\rangle$.

A _bounded morphism_ is a mapping $p:W\to V$ which preserves $R$ and satisfies
the _back condition_

$\displaystyle p(x)\mathrel{S}z\mathrel{\Rightarrow}(\exists y\in
W)[x\mathrel{R}y\text{ and }p(y)=z].$

If $p$ is onto we write $\langle
W,R\rangle\overset{b}{\twoheadrightarrow}\langle V,S\rangle$.

Suppose that $\mathfrak{F}$ is a class of frames. We call $\langle
V,S\rangle\in\mathfrak{F}$ _injective with respect to $\mathfrak{F}$_, if for
every injective bounded morphism $q:V\hookrightarrow W$ there is some
surjective bounded morphism $p:W\twoheadrightarrow V$ such that $p\circ q$ is
the identity on $W$.

If $R$ is a quasiorder, i.e. reflexive and transitive, then
$\theta_{R}:=\\{\langle x,y\rangle:xRy\text{ and }yRx\\}\cup 1^{\prime}$ is an
equivalence relation which can be partially ordered by
$x/\theta\leq_{R}y/\theta$ if and only if $x\mathrel{R}y$; the classes of
$\theta_{R}$ are called _clusters_.

Suppose that $\leq$ is a finite quasiorder on $W$. For $x\in W$ we let
$\mathop{\downarrow}x:=\\{y\in W:y\leq x\\}$, $\mathop{\uparrow}x:=\\{y\in
W:x\leq y\\}$, and $\mathop{\uparrow^{<}}x:=\\{y\in W:x\lneq y\\}$. If $x\lneq
y$, then _$y$ is a cover of $x$_ if $y$ is minimal in $\\{z\in W:x\lneq z\\}$.
$\emptyset\neq M\subseteq W$ is called an _antichain_ , if all elements of $M$
are pairwise incomparable. $M$ is called _dense_ (or _complete_), if each
element of $W$ is above some element of $M$. A _$\mu$ -set_ is a dense
antichain of $W$ [16]. We say that $\langle W,\leq\rangle$ is of _unification
type_

_nullary_ ,

if no $\mu$-set exists,

_unitary_ ,

if $W$ has a $\mu$ set, and every $\mu$-set has cardinality $1$,

_finitary_ ,

if each $\mu$-set is finite,

_infinitary_ ,

if it has an infinite $\mu$-set.

The _height_ $h(W)$ of $W$111Also called _depth_ or _rank_ [6, p. 46].  is the
length of a longest chain of clusters, and the _width_ $w(W)$ of $W$ is the
cardinality of the largest antichain; observe that the elements of an
antichain come from different clusters. The _local width_ $lw(W)$ of $W$ is
$\max\\{w(\mathop{\uparrow}x):x\in W\\}$, and the _covering width_ $cw(W)$ is
the largest number of covers any $x\in W$ has. Clearly, $cw(W)\leq lw(W)\leq
w(W)$. If $R$ is a rooted partial order, then $lw(W)=w(W)$, and, if $h(W)=2$,
then $c(w)=lw(W)$. However, in general none of the inequalities can be
reversed.

Throughout, $\langle B,+,\cdot,-,0,1\rangle$ is a non-trivial Boolean algebra,
usually referred to only by its universe. If $B$ is a subalgebra of a Boolean
Algebra $A$ we denote this by $B\leq A$. If $A$ is a homomorphic image of $B$
we write $B\twoheadrightarrow A$.

A _closure operator_ on $B$ is a mapping $f:B\to B$ which satisfies $f(0)=0$
and for all $a,b\in B$

$\displaystyle f(a+b)=f(a)+f(b),$ $\displaystyle a\leq f(a),$ $\displaystyle
f(f(a))\leq f(a).$

The _dual mapping_ of $f$ is defined by $f^{\partial}(a):=-f(-a)$ and denoted
by $f^{\partial}$; such operator is called _interior operator_. A _closure
algebra_ is a structure $\langle B,f\rangle$, where $B$ is a Boolean algebra
and $f$ is a closure operator; the pair $\langle B,f^{\partial}\rangle$ is
called an _interior algebra_. The variety of closure algebras is denoted by
$\mathbf{V_{Cl}}$. A closure algebra $\langle B,f\rangle$ is called _discrete_
, if $f$ is the identity. The two element closure algebra is denoted by
$\mathord{\mathbf{2}}$.

Throughout we suppose that $\mathbf{V}$ is a non-trivial variety of closure
algebras. $\mathbf{V}$ is _locally finite_ , if every finitely generated
$B\in\mathbf{V}$ is finite. An algebra $A\in\mathbf{V}$ is called _projective
in $\mathbf{V}$_ if and only if it is a retract of a free algebra.222Since
epimorphisms in $\mathbf{V_{Cl}}$ are onto [19] the notions of weak
projectivity of [3] and projectivity coincide. It is well known that $B$ is
projective in $\mathbf{V}$ if and only if for every $A\in\mathbf{V}$ and every
surjective homomorphism $p:A\twoheadrightarrow B$ there is some injective
homomorphism $q:B\hookrightarrow A$ such that $p\circ q$ is the identity on
$B$, in other words, if $B$ is a quotient of $A$, then it is a retract of $A$.

$\textstyle{{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{p\circ
q=1^{\prime}.}$

Note that $\mathord{\mathbf{2}}$ is projective in $\mathbf{V}$. If
$\mathbf{V}$ is locally finite the projectivity of a finite algebra depends
only on finite algebras:

######  Lemma 2.1.

Suppose that $\mathbf{V}$ is locally finite and that $B\in\mathbf{V}$ is
finite. Then $B$ is projective in $\mathbf{V}$ if and only if for every finite
$A\in\mathbf{V}$, every epimorphism $A\twoheadrightarrow B$ is a retraction.

###### Proof.

“$\Rightarrow$”: This is clear.

“$\Leftarrow$”: Suppose that $p:A\twoheadrightarrow B$ is an epimorphism. For
every $b\in B\setminus\\{0,1\\}$ choose some $a_{b}\in p^{-1}(b)$, and let
$A^{\prime}$ be the subalgebra of $A$ generated by $\\{a_{b}:b\in
B\setminus\\{0,1\\}\\}$. Since $\mathbf{V}$ is locally finite, $A^{\prime}$ is
finite. Clearly, $p^{\prime}:=p\mathrel{\upharpoonright}A^{\prime}$ is an
epimorphism $A^{\prime}\twoheadrightarrow B$. By the hypothesis, there is some
$q\hookrightarrow A^{\prime}\subseteq A$ such that $p^{\prime}\circ q=id(B)$,
and it follows from the choice of $A^{\prime}$ that $p\circ q=id(B)$. ∎

Let $\langle B,f\rangle$ be a nontrivial closure algebra. Its _canonical
frame_ is the structure $\operatorname{{\mathsf{Cf}}}(B):=\langle
W_{B},R_{f}\rangle$ where $W_{B}$ is the set of ultrafilters of $B$, and
$R_{f}$ is the binary relation on $W_{B}$ defined by

$F\mathrel{R_{f}}G$ if and only if $f[G]\subseteq F$.

Conversely, if $\mathfrak{W}:=\langle W,R\rangle$ is a frame, its _complex
algebra_ is the structure $\operatorname{{\mathsf{Cm}}}(\mathfrak{W}):=\langle
2^{W},\langle R\rangle\rangle$, where $\langle R\rangle:2^{W}\to 2^{W}$ is the
mapping defined by

$\displaystyle\langle R\rangle(X):=\\{x\in W:R(x)\cap X\neq\emptyset\\}.$

We denote $\operatorname{{\mathsf{Cm}}}\operatorname{{\mathsf{Cf}}}(B)$ by
$\operatorname{{\mathsf{Em}}}(B)$ and call it the _canonical embedding
algebra_ or _canonical extension_ of $B$. It is well known that the mapping
$h:B\to\operatorname{{\mathsf{Em}}}(B)$ is an embedding, and that
$B\cong\operatorname{{\mathsf{Em}}}(B)$ if and only if $B$ is finite [18]. It
is well known that $R_{f}$ is a quasiorder, and, if $R$ is a quasiorder, then
$\langle R\rangle$ is a closure operator [18]. The following facts are
decisive, see e.g. [4, Theorem 5.47]:

######  Lemma 2.2.

Suppose that $A,B$ are modal algebras and $\mathfrak{V},\mathfrak{W}$ are
frames. Then,

  1. 1.

If $A\hookrightarrow B$, then
$\operatorname{{\mathsf{Cf}}}(B)\overset{b}{\twoheadrightarrow}\operatorname{{\mathsf{Cf}}}(A)$.

  2. 2.

If $A\twoheadrightarrow B$, then
$\operatorname{{\mathsf{Cf}}}(B)\overset{g}{\hookrightarrow}\operatorname{{\mathsf{Cf}}}(A)$.

  3. 3.

If $\mathfrak{V}\overset{g}{\hookrightarrow}\mathfrak{W}$, then
$\operatorname{{\mathsf{Cm}}}(\mathfrak{W})\twoheadrightarrow\operatorname{{\mathsf{Cm}}}(\mathfrak{V})$.

  4. 4.

If $\mathfrak{V}\overset{b}{\twoheadrightarrow}\mathfrak{W}$, then
$\operatorname{{\mathsf{Cm}}}(\mathfrak{W})\hookrightarrow\operatorname{{\mathsf{Cm}}}(\mathfrak{V})$.

If all structures considered are finite, then
$\operatorname{{\mathsf{Em}}}(B)\cong B$,
$\operatorname{{\mathsf{Cf}}}\operatorname{{\mathsf{Cm}}}(\mathfrak{W})\cong\mathfrak{W}$,
and the implications above are, in fact, equivalences. Thus, in this situation
we may work either with frames or algebras. We shall use this duality
throughout without further reference.

##  3 Algebraic unification

Suppose that $\mathbf{V}$ is a locally finite variety of modal algebras. We
shall briefly describe the concept of algebraic unification as presented by
Ghilardi, [16] as applicable to locally finite varieties. Let $A\in\mathbf{V}$
be finite. A _unifier_ of $A$ is a pair $\langle u,B\rangle$ where $B$ is
finite and projective in $\mathbf{V}$, and $u:A\to B$ is a
homomorphism.333Strictly speaking we should define a unifier for
$A\in\mathbf{V}$ as a triple $\langle A,u,B\rangle$; we omit $A$ because we
consider only unifiers of a fixed $A$. Observe that projectivity, and
therefore, unifiability, depends on the chosen variety. The collection of
unifiers of $A$ is denoted by $U_{A}$.

Given two unifiers $\langle u,B\rangle$ and $\langle v,C\rangle$ of $A$, we
say that _$\langle u,B\rangle$ is more general than $\langle
v,C\rangle$_,444The quasiorder on $U_{A}$ is not uniformly defined in the
literature. We chose $\succcurlyeq$ to be consistent with [16] and the
$\mu$-sets of Section 2 . written as $\langle u,B\rangle\succcurlyeq\langle
v,C\rangle$, if there is a homomorphism $h:B\to C$ such that the diagram in
Figure 3 commutes, i.e. that $v=h\circ u$;

Figure 3: Quasiordering algebraic unifiers
$\textstyle{{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{v}$$\textstyle{{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\succcurlyeq}$$\textstyle{{C}}$

clearly, $\langle U_{A},\succcurlyeq\rangle$ is a quasiorder. If $A$ itself is
projective, then $\langle i,A\rangle$ is more general than any of its
unifiers. If $\langle u,B\rangle\preccurlyeq\langle v,C\rangle$ and $\langle
u,B\rangle\succcurlyeq\langle v,C\rangle$ we write $\langle
u,B\rangle\approx\langle v,C\rangle$, The relation $\approx$ is an equivalence
relation on $U_{A}$, and $U_{A}/\approx$ can be partially ordered as described
in Section 2. We say that the _unification type of $A$_ is the unification
type of the partially ordered set $\langle U_{A}/\approx\rangle$. Unification
of $A$ is called _filtering_ [17], if $U_{A}$ is up-directed: For every two
incomparable unifiers $\langle u_{1},B_{1}\rangle$ and $\langle
u_{2},B_{2}\rangle$ of $A$ there is a third unifier of $A$ more general than
both of them. Note that if the unification of $A$ is filtering, then the
unification type of $A$ is unitary or nullary.

By the homomorphism theorem a unifier $\langle u,B\rangle$ of $A$ is
determined by the closed ideal $u^{-1}(0)$ with associated congruence
$\theta=\ker(u)$, its canonical epimorphism $p_{\theta}:A\twoheadrightarrow
A/\theta$, and an embedding $e$ into $B$:

(3.5)
$\textstyle{{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\theta}}$$\scriptstyle{u}$$\textstyle{{A/\theta}~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{{B}}$

In this sense, we can think of a unifier of $A$ as a triple
$\langle\theta,e,B\rangle$ where $\theta$ is a congruence on $A$, $B$ is a
finite algebra projective in $\mathbf{V}$, and $e:A\hookrightarrow B$ is an
embedding. Note that $A/\theta$ need not be projective, but only needs to be
embeddable into a projective algebra, equivalently, into a free algebra. If
$A/\theta$ is itself projective, then $\langle p_{\theta},A/\theta\rangle$ is
a unifier of $A$, more general than $\langle u,B\rangle$.

We denote by $\mathbb{C}(A)$ the set of congruences of $A$ for which
$A/\theta$ is isomorphic to a subalgebra of some projective algebra, also
called _admissible congruences_. These congruences are decisive when
investigating unifiers.

Using the decomposition of $u$ and $v$ as outlined above, we arrive at the
situation in Figure 4.

Figure 4: Quasiordering algebraic unifiers using quotients
$\textstyle{{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\theta}}$$\scriptstyle{p_{\psi}}$$\textstyle{A/\theta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{\theta}}$$\textstyle{A/\psi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{\psi}}$$\textstyle{{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\succcurlyeq}$$\textstyle{{C}}$

Below we collect some simple properties of algebraic unifiers which we shall
use later on.

######  Lemma 3.1.

  1. 1.

Suppose that $\langle u,B\rangle,\langle v,C\rangle$ are unifiers of $A$. If
$\langle u,B\rangle\succcurlyeq\langle v,C\rangle$, then
$\ker(u)\subseteq\ker(v)$. Consequently,

    1. (a)

If $\ker(u)$ and $\ker(v)$ are incomparable with respect to $\subseteq$, then
$\langle u,B\rangle$ and $\langle v,C\rangle$ are incomparable with respect to
$\succcurlyeq$.

    2. (b)

$\langle u,B\rangle\approx\langle v,C\rangle$ implies $\ker(u)=\ker(v)$.

  2. 2.

If $\langle u,B\rangle\in U_{A}$, $C$ is projective in $\mathbf{V}$, and $B$
is a retract of $C$, then, $\langle u,B\rangle\approx\langle u,C\rangle$.

###### Proof.

1\. Suppose that $h:B\to C$ with $h\circ u=v$. If $u(x)=0$, then
$h(u(x))=v(x)=0$.

2\. Suppose w.l.o.g. that $B\leq C$ with $i:B\hookrightarrow C$ the identity
embedding, and that $p:C\twoheadrightarrow B$ is a retraction:

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{i\circ
u}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$

Since $i(u(x))=u(x)$ we have $\langle u,B\rangle\succcurlyeq\langle
u,C\rangle$. For the converse, let $x\in A$; then, $u(x)\in B\leq C$, and
$p(i(u(x)))=u(x)$, since $p\mathrel{\upharpoonright}B$ is the identity. ∎

######  Corollary 3.2.

  1. 1.

Suppose that $\langle u,B\rangle$ is a unifier of $A$. Then, there is a finite
free algebra $F$ such that $\langle u,B\rangle\approx\langle u,F\rangle$.

  2. 2.

If $\langle u,\mathord{\mathbf{2}}\rangle$ is a unifier of $A$ and $B$ is
finite and projective in $\mathbf{V}$, then $\langle
u,\mathord{\mathbf{2}}\rangle\approx\langle u,B\rangle$.

###### Proof.

1\. Since $B$ is projective, it is a retract of a free algebra $F$. Since
$u[A]$ is finite, we may assume $F$ to be finite as well.

2\. By the hypothesis, $\langle u,\mathord{\mathbf{2}}\rangle$ is a unifier of
$A$. Since $B$ is projective, $\mathord{\mathbf{2}}$ is a retract of $B$, and
the claim follows from Lemma 3.1(2). ∎

Even $u[A]\cong v[A]$ does not imply that $\langle
u,B\rangle\succcurlyeq\langle v,B\rangle$:

######  Example 3.1.

Let $A\in\mathbf{V}$ and $F,G$ be different closed prime ideals of $A$, and
$p_{F}:A\twoheadrightarrow A/F$ and $p_{G}:A\twoheadrightarrow A/G$ be the
canonical epimorphisms; then, $A/F=A/G=\mathord{\mathbf{2}}$. If $a\in
F\setminus G$, then $0=p_{F}(a)\neq p_{G}(a)=1$ which shows that there is no
homomorphism $h:\mathord{\mathbf{2}}\to\mathord{\mathbf{2}}$ such that $h\circ
p_{F}=p_{G}$. ∎

######  Lemma 3.3.

[8, p. 50] If $f:A\twoheadrightarrow B$ is surjective, and $g:A\to C$ is a
homomorphism such that $\ker(f)\subseteq\ker(g)$, there is some $h:B\to C$
such that $g=h\circ f$.

######  Lemma 3.4.

Suppose that $\langle u,B\rangle$ is a unifier of $A$ and $h:B\to C$ is an
isomorphism. Then, $\langle u,B\rangle\approx\langle h\circ u,C\rangle$.

###### Proof.

Just consider the diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{h\circ
u}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{-1}}$

∎

##  4 The fork

The _$2$ -pronged fork_, or simply _fork_ is the frame $F$ shown in Figure 2.
Its complex algebra is denoted by $B_{F}$, and the variety it generates by
$\operatorname{\mathbf{Eq}}(B_{F})$; since $\operatorname{\mathbf{Eq}}(B_{F})$
is a finite variety, it is locally finite. A _fork algebra_ is a nontrivial
finite algebra in $\operatorname{\mathbf{Eq}}(B_{F})$.555These should not be
confused with the fork algebras of e.g. [14]. A _fork frame_ has the form
$\operatorname{{\mathsf{Cf}}}(B)$ for a fork algebra $B$. Aiello et al., [1,
Theorem 5.7] have shown that the variety $\operatorname{\mathbf{Eq}}(B_{F})$
is the variety of closure algebras which is characterized by the axioms

(4.1) $\displaystyle f^{\partial}(f(x\cdot f(-x))+x)\leq x$
$\displaystyle\mathbf{Grz},$ (4.2) $\displaystyle-x\cdot f(x)\leq
f(f^{\partial}(x))$ $\displaystyle\mathbf{BD_{2}},$ (4.3)
$\displaystyle-(x\cdot y\cdot f(x\cdot-y)\cdot f(-x\cdot y)\cdot
f(-x\cdot-y))=1$ $\displaystyle\mathbf{BW_{2}}$

The axiom Grz implies that a (finite) fork frame is a partial order [6], which
we will denote by $\leq$, possibly indexed. The first order frame conditions
corresponding to (4.2) , respectively, to (4.3) are 666Computed by SQEMA [15].

(4.4) $\displaystyle\forall y(x\leq y\mathrel{\Rightarrow}(x=y\text{ or
}\exists z_{1}(x\leq z_{1}\text{ and }\forall z_{2}(z_{1}\leq
z_{2}\mathrel{\Rightarrow}y=z_{2}))))$

and

(4.5) $\forall y_{1}\forall y_{2}((x\leq y_{1}\text{ and }x\leq
y_{2})\mathrel{\Rightarrow}(x=y_{1}\text{ or }x=y_{2}\text{ or
}y_{1}=y_{2}\text{ or }\\\ \forall z_{1}(x\leq
z_{1}\mathrel{\Rightarrow}(x=z_{1}\text{ or }y_{1}=z_{1}\text{ or
}y_{2}=z_{1})))).$

Together, (4.2) says that the height of a fork frame is at most two, and (4.3)
says that every $x$ is related to at most two other elements. Together they
imply that a rooted fork frame has one of the following forms:

---
$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

We denote by $L_{W}^{1}$ the points on the lower level and by $L_{W}^{2}$ the
points on the upper level of a fork frame $W$. The points in $L_{W}^{1}$
correspond to the closed atoms of $\operatorname{{\mathsf{Cm}}}(W)$, whereas
$L_{W}^{2}$ corresponds to the set of its non-closed atoms.

##  5 Finite projective fork algebras

Our first result gives a necessary condition for a finite
$B\in\operatorname{\mathbf{Eq}}(B_{F})$ to be projective. We prove a more
general result:

######  Theorem 5.1.

Let $F_{n,m}$ be the class of finite partial orders of bounded height $n\geq
2$ and covering width $m\geq 2$, and $\mathbf{V}_{n,m}$ be its associated
variety. Suppose that $B\in\mathbf{V}_{n,m}$ is finite and projective. Then,
$B$ is directly indecomposable, and
$\mathop{\downarrow}f(a_{1})\cap\ldots\cap\mathop{\downarrow}f(a_{k})\neq\\{0\\}$
for all non-closed atoms $a_{1},\ldots a_{k}$, when $k\leq m$.

###### Proof.

Suppose that $\langle V,\leq_{V}\rangle$ is the canonical frame of $B$; we
shall prove the dual statement. Assume that there are $v_{1},\ldots,v_{k}\in
L_{V}^{2}$ such that
$\mathop{\downarrow}v_{1}\cap\ldots\cap\mathop{\downarrow}v_{k}=\emptyset$.
Choose some $x\not\in V$, and set $W:=V\cup\\{x\\}$,
$\leq_{W}:=\leq_{V}\cup\\{\langle x,v_{i}\rangle:1\leq i\leq
k\\}\cup\\{\langle x,x\rangle\\}$; then, $\mathop{\uparrow}_{W}{V}\subseteq
V$, and therefore $V$ is a generated substructure of $W$. Since the height of
$V$ is at least two, adding $x$ as above does not increase the height, hence,
$\operatorname{{\mathsf{Cm}}}(W)\in\mathbf{V}_{n,m}$. Suppose that
$p:W\twoheadrightarrow V$ is a bounded retraction, and $p(x)=y$. Since $p$
preserves order, $y\leq v_{i}=p(v_{i})$ for $1\leq i\leq k$. This contradicts
the hypothesis.

Assume that $V_{1},V_{2}$ are different connected components of $V$, and let
$y_{i}\in V_{i}$ be maximal. Choose some $x\not\in V$, and set
$W:=V\cup\\{x\\}$, $\leq_{W}:=\leq_{V}\cup\\{\langle x,x\rangle,\langle
x,y_{1}\rangle,\langle x,y_{2}\rangle\\}$. Since $n,m\geq 2$, height and cover
width of $V$ are not increased, and we can proceed as above to arrive at a
contradiction. ∎

It follows that a finite algebra projective in $\mathbf{V}_{n,m}$ is not a
direct product of projective algebras is projective. Therefore, in
$\mathbf{V}_{n,m}$ unification is not filtering in the sense of Ghilardi and
Sacchetti, [17, Theorem 3.2]. Since
$\operatorname{\mathbf{Eq}}(B_{F})=\mathbf{V}_{2,2}$, we obtain

######  Corollary 5.2.

If $\langle B,f\rangle\in\operatorname{\mathbf{Eq}}(B_{F})$ is finite and
projective, then $B$ is directly indecomposable and
$\mathop{\downarrow}f(x)\cap\mathop{\downarrow}(f(y))\neq\\{0\\}$ for all non-
closed atoms $x,y\in B$.

We now show that the conditions of Corollary 5.2 are sufficient for
projectivity.

######  Theorem 5.3.

Suppose that $\langle B,f\rangle\in\operatorname{\mathbf{Eq}}(B_{F})$ is
finite and directly indecomposable, and that
$\mathop{\downarrow}f(x)\cap\mathop{\downarrow}f(y)\neq\\{0\\}$ for all non-
closed atoms $x,y\in B$. Then, $B$ is projective in
$\operatorname{\mathbf{Eq}}(B_{F})$.

###### Proof.

Suppose w.l.o.g. that $B\neq\mathord{\mathbf{2}}$. We will use duality, and
set $\langle V,\leq\rangle:=\operatorname{{\mathsf{Cf}}}(B)$; furthermore, we
suppose that $V$ is a generated substructure of a fork frame $W$. Since $V$ is
connected, it is contained in a component of $W$, and by mapping all points of
$W$ outside this component to a maximal point of $V$, we may suppose w.l.o.g.
that $W$ itself is connected; in particular, for all $x\in L_{W}^{1}$ there is
some $y\in L_{W}^{2}$ such that $x\leq y$.

We will construct a bounded epimorphism $p:W\twoheadrightarrow V$ by cases. It
suffices to show that $p$ preserves $\leq$ and satisfies the back condition on
$\mathop{\uparrow}x$ for $x\in L_{W}^{1}$. Let $p$ be the identity on $V$. We
divide $L_{W}^{1}\setminus V$ into three disjoint (possibly empty) sets:

(5.1) $\displaystyle W_{1}$ $\displaystyle:=\\{x\in L_{W}^{1}\setminus
V:\mathop{\uparrow^{<}}x\subseteq V\\},$ (5.2) $\displaystyle W_{2}$
$\displaystyle:=\\{x\in L_{W}^{1}\setminus V:\mathop{\uparrow^{<}}x\cap
V\neq\emptyset,\mathop{\uparrow^{<}}x\cap W\setminus V)\neq\emptyset\\},$
(5.3) $\displaystyle W_{3}$ $\displaystyle:=\\{x\in L_{W}^{1}\setminus
V:\mathop{\uparrow^{<}}x\cap V=\emptyset\\}.$

  1. 1.

$x\in W_{1}$: Here, we consider two cases:

    1. (a)

$\mathop{\uparrow^{<}}x=\\{v\\}$: Set $p(x):=v$. Then,
$p[\mathop{\uparrow}x]=\\{v\\}$ and clearly,
$p\mathrel{\upharpoonright}\mathop{\uparrow}x$ is a bounded morphism.

    2. (b)

$\mathop{\uparrow^{<}}x=\\{v,w\\}$, $v\neq w$: Choose some
$u\in\mathop{\downarrow}v\cap\mathop{\downarrow}w\cap V$, and set $p(x):=u$;
such $u$ exists by the hypothesis. Then, the diagram

---
$\textstyle{v}$$\textstyle{w}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{u\in
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

shows that $p\mathrel{\upharpoonright}[\mathop{\uparrow}x]$ is a bounded
morphism.

  2. 2.

$x\in W_{2}$: Define

$\displaystyle Y:=\bigcup\\{\mathop{\uparrow}x:x\in
W_{2}\\}\cap(L_{W}^{2}\setminus V);$

then,

$\displaystyle Y=\\{y\in L_{W}^{2}\setminus V:(\exists x\in L_{W}^{1})[x\leq
y\text{ and }\mathop{\uparrow^{<}}x\cap L_{V}^{2}\neq\emptyset]\\}.$

For each $y\in Y$ we set

$\displaystyle X_{y}:=\\{x\in L_{W}^{1}:x\leq y\text{ and
}\mathop{\uparrow}x\cap L_{V}^{2}\neq\emptyset\\}.$

The situation $x\in X_{y}$ is depicted in the following diagram:

---
$\textstyle{y}$$\textstyle{v}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

If $x\in X_{y}$, then $x\not\in V$, since $x\leq y\not\in V$ and $V$ is a
generated substructure of $W$.

Our next aim is to show that $\\{X_{y}:y\in Y\\}$ is a partition of $W_{2}$.
Assume that $y,y^{\prime}\in Y,y\neq y^{\prime}$, and $x\in X_{y}\cap
X_{y^{\prime}}$. Then, $x\lneq y,x\lneq y^{\prime}$ and therefore,
$\mathop{\uparrow^{<}}x=\\{y,y^{\prime}\\}\subseteq L_{W}^{2}\setminus V$ by
(4.3), respectively, (4.5). This contradicts $x\in W_{2}$. If $x\in W_{2}$
there is some $y\in Y$ such that $\leq y$, thus, $x\in X_{y}$. Hence,
$p\mathrel{\upharpoonright}X_{y}$ and
$p\mathrel{\upharpoonright}X_{y^{\prime}}$ may be defined independently if
$y\neq y^{\prime}$.

Let $y\in Y$ and enumerate $X_{y}=\\{x_{1},\ldots,x_{k}\\}$; then, $x_{i}\leq
y$ and $x_{i}\leq v_{i}$ for exactly one $v_{i}\in L_{V}^{2}$; note that the
$v_{i}$ are not necessarily different. Suppose w.l.o.g. that $v_{i}=v_{1}$ for
$1\leq i\leq m\leq k$, and set $p(y):=v_{1}$ as well as $p(x_{i}):=v_{1}$ for
$1\leq i\leq m$.

$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{v_{1}=v_{2}}$$\textstyle{x_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$.$\textstyle{x_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$.

Then, $p[\bigcup\\{\mathop{\uparrow}x_{i}:1\leq i\leq m\\}]=\\{v_{1}\\}$, and
clearly, $p\mathrel{\upharpoonright}\bigcup\\{\mathop{\uparrow}x_{i}:1\leq
i\leq m\\}$ is a bounded morphism.

For $i=m+1,\ldots,k$ choose some $u_{i}\in L_{V}^{1}$ such that
$u_{i}\in\mathop{\downarrow}v_{1}\cap\mathop{\downarrow}v_{i}$ and define
$p(x_{i}):=u_{i}$. Now, the diagram

$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{v_{1}}$$\textstyle{v_{i}}$$\textstyle{x_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{u_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

shows that $p\mathrel{\upharpoonright}\mathop{\uparrow}x_{i}$ is a bounded
morphism. This way we have defined
$p\mathrel{\upharpoonright}\bigcup\\{\mathop{\uparrow}x:x\in W_{2}\\}$.

  3. 3.

$x\in W_{3}$: Thus far, we have well defined $p$ on $V$ and
$\mathop{\uparrow}(W_{1}\cup W_{2})$. If
$x\in\mathop{\downarrow}y\cap\mathop{\downarrow}y^{\prime}$ for some distinct
$y,y^{\prime}\in L_{W}^{2}\setminus V$, then $p$ might already been defined on
$\mathop{\uparrow^{<}}x$ in the previous step for example,

---
$\textstyle{y}$$\textstyle{y^{\prime}}$$\textstyle{v=p(y^{\prime})}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

First, suppose that $\mathop{\uparrow^{<}}x=\\{y\\}$. If $p(y)$ has already
been defined, set $p(x):=p(y)$. Otherwise, choose some $v\in L_{V}^{2}$ and
set $p(x):=v,p(y):=v$.

Finally, let $\mathop{\uparrow^{<}}x=\\{y,z\\}$, $y\neq z$. If both $p(y)$ and
$p(z)$ have been defined choose
$u\in\mathop{\downarrow}p(y)\cap\mathop{\downarrow}p(z)\cap V$ and set
$p(x):=u$. Then,

---
$\textstyle{y}$$\textstyle{z}$$\textstyle{p(y)}$$\textstyle{p(z)}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{u\in
V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

shows that $p\mathrel{\upharpoonright}\mathop{\uparrow}x$ is a bounded
morphism.

If only one of $p(y),p(z)$ has been defined, say, $p(y)$, choose $v\in
L_{V}^{2}$, set $p(z):=v$, choose
$u\in\mathop{\downarrow}p(y)\cap\mathop{\downarrow}v$ and set $p(x):=u$:

---
$\textstyle{y}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{v}$$\textstyle{p(y)}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{u\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

As in the previous case, $p\mathrel{\upharpoonright}\mathop{\uparrow}x$ is a
bounded morphism. If neither $p(y)$ nor $p(z)$ have been defined, choose $u\in
L_{V}^{1},v\in L_{V}^{2}$ such that $u\lneq v$, and set $p(y),p(z):=v$ and
$p(x):=u$. Clearly, $p\mathrel{\upharpoonright}\mathop{\uparrow}x$ is an
epimorphism.

This completes the proof. ∎

######  Corollary 5.4.

If $A$ is a finite directly indecomposable fork algebra with at most two non-
closed atoms, then $A$ is projective.

###### Proof.

If $A$ has only one non-closed atom, then $A$ is projective by Theorem 5.3.
Thus, suppose that $v,w$ are the non-closed atoms of $A$. Let $\langle
V,\leq\rangle$ be the canonical frame of $A$; then, $L_{V}^{2}=\\{v,w\\}$.
Since $A$ is directly indecomposable, $V$ is connected, hence, there is a path
from $v$ to $w$. The restriction of $\leq$ to the levels is the identity and
therefore, there is some $u\in L_{V}^{1}$ such that $u\leq v$ and $u\leq w$.
Thus, the conditions of Theorem 5.3 are met, hence, $A$ is projective ∎

The variety of all closure algebras has no non-trivial injectives [5, Theorem
7.12]. The situation is different for $\operatorname{\mathbf{Eq}}(B_{F})$:

######  Theorem 5.5.

$B_{F}$ is injective in $\operatorname{\mathbf{Eq}}(B_{F})$.

###### Proof.

We will prove the dual statement and use the fork as in Figure 2. Let $W$ be a
fork frame, $p:W\twoheadrightarrow F$ be a bounded epimorphism, and suppose
that $p(x)=u$. Since $p$ is bounded, there are $y,z\in W$ such that $x\leq
y,x\leq z$ and $p(y)=v,p(z)=w$. Then, the assignment $u\mapsto x,v\mapsto y$
and $w\mapsto z$ shows that $p$ is a retraction. It remains to show that
$\\{x,y,z\\}$ is a generated substructure of $W$, i.e. that
$\mathop{\uparrow}x=\\{x,y,z\\}$. This follows immediately from the height and
width conditions (4.4) and (4.5). ∎

##  6 The unification type of the variety generated by the fork

It was shown by Dzik et al., [12, Corollary 4.8] that
$\operatorname{\mathbf{Eq}}(B_{F})$ has finitary but not unitary unification.
In this section we will give a purely semantic proof for this result.
Throughout this section $\langle
A,f\rangle\in\operatorname{\mathbf{Eq}}(B_{F})$ is finite. If $\langle
u,B\rangle$ is a unifier of $A$ we may suppose that $u=e\circ p$ for some
admissible congruence $\theta$ on $A$ and $p:A\twoheadrightarrow A/\theta$
such that $p[A]\cong u[A]\leq B$, and $e:p[A]\hookrightarrow B$ is an
embedding.

Consider the fork frame $W$ pictured in Figure 5.

Figure 5: The $W$ frame
---
$\textstyle{t}$$\textstyle{v}$$\textstyle{w}$$\textstyle{u\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{u^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The complex algebra of $W$ is denoted by $\langle B_{W},f_{W}\rangle$. It has
5 atoms, and we identify these with the points of $W$. The action of $f_{W}$
on the atoms is given by

$\begin{array}[]{l|ccccc}x&u&u^{\prime}&t&v&w\\\
f_{W}(x)&u&u^{\prime}&u+t&u+u^{\prime}+v&u^{\prime}+w\end{array}$

The first result in this section establishes a connection between $B_{W}$ and
projective fork algebras.

######  Theorem 6.1.

If $A\in\operatorname{\mathbf{Eq}}(B_{F})$ is finite and directly
indecomposable, then $A$ is projective in $\operatorname{\mathbf{Eq}}(B_{F})$
if and only if $B_{W}$ is not a subalgebra of $A$.

###### Proof.

“$\Rightarrow$”: We use duality and the notation of $W$ as above. Suppose that
$V$ is an injective fork frame and assume that $p:V\twoheadrightarrow W$ is a
bounded epimorphism. Let $r,s\in V$ such that $p(x)=u$ and $p(y)=u^{\prime}$.
Since $p$ is bounded there are $r,s\in L_{V}^{2}$ such that $x\leq r,y\leq s$
and $p(r)=t,p(s)=w$. Since $V$ is injective, there is some $z\in L_{V}^{1}$
such that $z\leq r$ and $z\leq s$. Since $p$ preserves order we have $p(z)\leq
p(r)=t$ and $p(z)\leq p(s)=w$. However, since
$\mathop{\downarrow}t\cap\mathop{\downarrow}w=\emptyset$, such $z$ does not
exist.

“$\Leftarrow$”: Suppose that $B_{W}\leq A$; we need to find two non-closed
atoms $a,a^{\prime}$ of $A$ with $f(a)\cdot f(a^{\prime})=0$. With some abuse
of language we let the atoms of $B_{W}$ be the points in Figure 5; then,
$u=f(u)\leq f(t)\cdot f(v)$, $u^{\prime}=f(u^{\prime})\leq f(w)\cdot f(v)$ and
$f(v)\cdot f(w)=0$. Let $a_{t}$ and $a_{w}$ be non-closed atoms of $A$ with
$a_{t}\leq t,a_{w}\leq w$. These exist, since each atom of $B_{W}$ is a sum of
atoms of $A$, and $t,w$ are non-closed. Now, $a\neq f(a)\leq
f(t),a^{\prime}\neq f(a^{\prime})\leq f(w)$ and therefore $f(a)\cdot
f(a^{\prime})=0$. It follows that $A$ is not projective by Theorem 5.1. ∎

######  Theorem 6.2.

The unification type of $\operatorname{\mathbf{Eq}}(B_{F})$ is finitary and
not unitary.

###### Proof.

Suppose that $A\in\operatorname{\mathbf{Eq}}(B_{F})$ is finite, and
$\theta\in\mathbb{C}(A)$; then, there is some projective
$B\in\operatorname{\mathbf{Eq}}(B_{F})$ such that $A/\theta\leq B$. Since
$\operatorname{\mathbf{Eq}}(B_{F})$ is locally finite, we may suppose that $B$
is finite as well. Furthermore, $A/\theta$ is directly indecomposable, since
it is a subalgebra of the projective algebra $B$ which is directly
indecomposable.

Our first aim is to show that $A/\theta$ itself is projective. Assume
otherwise; then $B_{W}$ is a subalgebra of $A/\theta$ by Theorem 6.1. Since
$A/\theta$ is a subalgebra of $B$, the algebra $B_{W}$ is also a subalgebra of
$B$, contradicting that $B$ is projective. This shows that $\langle
p_{\theta},A/\theta\rangle$ is a unifier of $A$ for an admissible congruence
$\theta$.

Suppose that $\langle u,B\rangle$ is a unifier of $A$. Then, there are some
admissible $\theta$ such that $p_{\theta}[A]=u[A]$, and some embedding
$e:p_{\theta}[A]\hookrightarrow B$ such that $u=e\circ p_{\theta}$. Hence,
$\langle p_{\theta},p_{\theta}[A]\rangle\succcurlyeq\langle u,B\rangle$:

$\textstyle{{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\theta}}$$\scriptstyle{u=e\circ
p_{\theta}}$$\textstyle{{A/\theta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{\succcurlyeq}$$\textstyle{{B}}$

Thus, each unifier is below a unifier of the form $\langle
p_{\theta},p_{\theta}[A]\rangle$ for some $\theta\in\mathbb{C}(A)$. As
$\mathbb{C}(A)$ is finite, there can be no infinite $\mu$ set.

Let $M$ be a maximal antichain in $\mathbb{C}(A)$. Then, $\\{\langle
p_{\theta},p_{\theta}[A]\rangle:\theta\in M\\}$ is a $\mu$ set: Let $\langle
u,B\rangle\in U_{A}$ and $u=e\circ p_{\theta}$; then, $\theta\in\mathbb{C}(A)$
and $\langle p_{\theta},p_{\theta}[A]\rangle\succcurlyeq\langle u,B\rangle$.
The maximality of $M$ now implies that $\langle
p_{\psi},p_{\psi}[A]\rangle\succcurlyeq\langle
p_{\theta},p_{\theta}[A]\rangle$ for some $\psi\in M$.

Thus far we have shown that the unification type of
$\operatorname{\mathbf{Eq}}(B_{F})$ is neither infinitary nor nullary. It
remains show that unification in $\operatorname{\mathbf{Eq}}(B_{F})$ is not
unitary. This follows from the more general Theorem 6.5. ∎

We continue by exhibiting the unifier classes of $B_{W}$, thus presenting a
concrete example of an algebra in $\operatorname{\mathbf{Eq}}(B_{F})$ without
unitary unification.

######  Example 6.1.

If $a\in B_{W}$ is closed, we let $p_{a}:B_{W}\twoheadrightarrow
B_{W}/\mathop{\downarrow}a$ be the canonical epimorphism. An
inspection777Obtained from [13]. The source is available from the first
author. of the quotients of $B_{W}$ shows that of the 11 proper non-identity
congruences of $B_{W}$ only 5 lead to directly indecomposable algebras. Since
projective algebras in $\operatorname{\mathbf{Eq}}(B_{F})$ are directly
indecomposable and direct indecomposability is a universal property, these are
the only ones which may appear as subalgebras of a projective algebra. Three
of these are isomorphic to $\mathord{\mathbf{2}}$, namely,
$B_{W}/\mathop{\downarrow}-t,B_{W}/\mathop{\downarrow}-v$ and
$B_{W}/\mathop{\downarrow}-w$, leading to the incomparable unifiers $\langle
p_{-t},B_{W}/\mathop{\downarrow}-t\rangle,\langle
p_{-v},B_{W}/\mathop{\downarrow}-v\rangle,\langle
p_{-w},B_{W}/\mathop{\downarrow}-w\rangle$. Since prime ideals are
incomparable, it follows that these unifiers are incomparable with respect to
$\succcurlyeq$ by Lemma 3.1. The other two directly indecomposable quotients
are $B_{W}^{1}:=B_{W}/\mathop{\downarrow}f(t)$ and
$B_{W}^{2}:=B_{W}/\mathop{\downarrow}f(w)$, each of which is isomorphic to the
projective algebra $B_{F}$. Let $p_{i}:B_{W}\twoheadrightarrow B_{W}^{i}$ be
the respective quotient mappings; then, $\langle p_{1},B_{W}^{1}\rangle$ and
$\langle p_{2},B_{W}^{2}\rangle$ are unifiers of $B_{W}$. In what follows we
will show the following:

  1. 1.

The unifiers $\langle p_{1},B_{W}^{1}\rangle$ and $\langle
p_{2},B_{W}^{2}\rangle$ are incomparable with respect to $\succcurlyeq$.

  2. 2.

Every unifier of $B_{W}$ with at least four elements is equivalent to $\langle
p_{1},B_{W}^{1}\rangle$ or $\langle p_{2},B_{W}^{2}\rangle$.

1\. follows immediately from Lemma 3.1 since
$\mathop{\downarrow}f(t)\cap\mathop{\downarrow}f(w)=\\{0\\}$.

For 2., suppose that $\langle u,A\rangle\in U_{B_{W}}$ has at least four
elements. Since $A$ is projective, $B_{W}$ is not isomorphic to a subalgebra
of $A$ by Theorem 6.1, and therefore $\ker(u)=\mathop{\downarrow}f(t)$ or
$\ker(u)=\mathop{\downarrow}f(w)$, apart from those quotients isomorphic to
$\mathord{\mathbf{2}}$. Suppose w.l.o.g. that
$\ker{u}=\mathop{\downarrow}f(t)$. Since $\ker(p_{1})=\ker(u)$, there is an
isomorphism $q:u[B_{W}]\to B_{W}^{1}$ such that $q\circ u=p_{1}$. Furthermore,
$u[B_{W}]\cong B_{F}$, so there is a retraction $g:A\twoheadrightarrow
u[B_{W}]$ by Theorem 5.5. We now have the following situation, where all
diagrams commute:

---
$\textstyle{B_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{p_{1}}$$\scriptstyle{u}$$\textstyle{u[B_{W}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{q}$$\textstyle{B_{W}^{1}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{q\circ
g}$

Set $h:=q\circ g$ and let $x\in B_{W}$. Then, noting that
$g\mathrel{\upharpoonright}u[B_{W}]$ is the identity, we obtain

$\displaystyle(h\circ u)(x)=(q\circ g\circ u)(x)=(q\circ u)(x)=p_{1}(x),$

which shows that $\langle u,A\rangle\succcurlyeq\langle
p_{1},B_{W}^{1}\rangle$. Next, we show that $\langle
p_{1},B_{W}^{1}\rangle\succcurlyeq\langle u,A\rangle$. Let $q$ be as above and
consider the following diagram:

$\textstyle{B_{W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{p_{1}}$$\textstyle{u[B_{W}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{B_{W}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{q^{-1}}$$\textstyle{A}$

Then, $q^{-1}\circ p_{1}:B_{W}\to A$, and $(q^{-1}\circ p_{1})(x)=(q^{-1}\circ
q\circ u)(x)=u(x)$ shows that $\langle
p_{1},B_{W}^{1}\rangle\succcurlyeq\langle u,A\rangle$. Finally, $\langle
u,A\rangle\not\succcurlyeq\langle p_{2},B_{W}^{2}\rangle$, since $\ker(p_{1})$
and $\ker(p_{2})$ are incomparable. Thus, both our claims are proved. ∎

Next, we extend Theorem 6.2 to show that no variety $\mathbf{V}$ of closure
algebras containing $B_{F}$ has unitary unification and prove a converse
relative to varieties with finitary unification. First, we recall the concept
of a splitting pair of a lattice, see e.g. [5]. A pair
$\langle\mathbf{V}_{1},\mathbf{V}_{2}\rangle$ of subvarieties of
$\mathbf{V_{Cl}}$ is called _splitting_ , if

  1. 1.

$\mathbf{V}_{1}\not\leq\mathbf{V}_{2}$,

  2. 2.

If $\mathbf{V}^{\prime}\leq\mathbf{V_{Cl}}$, then
$\mathbf{V}_{1}\leq\mathbf{V}^{\prime}$ or
$\mathbf{V}^{\prime}\leq\mathbf{V}_{2}$.

It is well known that $\mathbf{V}_{1}$ is generated by a finite subdirectly
irreducible algebra, say, $B$, and that $\mathbf{V}_{2}$ is the largest
subvariety of $\mathbf{V_{Cl}}$ not containing $B$, called the _splitting
companion of $B$_; for details see e.g. [5]. It has been known for some time
[20, Example, p. 158] that the splitting companion of $B_{F}$ is the variety
$\mathbf{V}_{G}$ of closure algebras $\langle B,f\rangle$ which satisfy the
Geach condition

  1. G

$f(f^{\partial}(x))\leq f^{\partial}(f(x))$.

These algebras are the algebraic models of the logic S4.2.

We say that $\mathbf{V}$ satisfies the _weak disjunction property_ (WDP) [10]
if for all terms $\tau_{1},\tau_{2}$ in the language of $\mathbf{V}$,

  1. WDP

$\mathbf{V}\models f^{\partial}(\tau_{1})+f^{\partial}(\tau_{2})\approx
1\mathrel{\Rightarrow}\operatorname{\mathbf{Eq}}(\mathord{\mathbf{2}})\models\tau_{1}\approx
1\text{ or
}\operatorname{\mathbf{Eq}}(\mathord{\mathbf{2}})\models\tau_{2}\approx 1.$

It turns out that the WDP is intimately related to
$\operatorname{\mathbf{Eq}}(B_{F})$:

######  Theorem 6.3.

$\mathbf{V}$ satisfies the WDP if and only if
$\operatorname{\mathbf{Eq}}(B_{F})\leq\mathbf{V}$.

###### Proof.

“$\Rightarrow$”: Suppose that $\mathbf{V}\leq\mathbf{V_{Cl}}$ satisfies the
_WDP_ , and assume that $\operatorname{\mathbf{Eq}}(B_{F})\not\leq\mathbf{V}$.
Then $\mathbf{V}\leq\mathbf{V}_{G}$, and thus, $A\models
f^{\partial}(f(-x)))+f^{\partial}(f(x))\approx 1$ for all $A\in\mathbf{V}$.
Choose some $A\in\mathbf{V},A\neq\mathord{\mathbf{2}}$, and some $x\in
A,x\not\in\\{0,1\\}$. Since $\mathbf{V}$ satisfies the _WDP_ , $A^{B}\models
x\approx 1$ or $A^{B}\models-x\approx 1$, where $A^{B}$ is the Boolean reduct
of $A$. This contradicts the assumption $x\not\in\\{0,1\\}$.

“$\Leftarrow$”: This is [10, Lemma 10] . ∎

Unification is related to the WDP by the following result:

######  Lemma 6.4.

[10, Lemma 9] If $\mathbf{V}$ satisfies the WDP, it does not have unitary
unification.

######  Theorem 6.5.

  1. 1.

If $B_{F}\in\mathbf{V}$, then $\mathbf{V}$ does not have unitary unification,
equivalently, if $\mathbf{V}$ has unitary unification, then
$B_{F}\notin\mathbf{V}$.

  2. 2.

If $\mathbf{V}$ has finitary unification and not unitary unification, then
$B_{F}\in\mathbf{V}$.

###### Proof.

1\. This follows from Theorem 6.3 and Lemma 6.4.

2\. Suppose that $\mathbf{V}$ has finitary unification and not unitary
unification. Then, there is some $\langle A,f\rangle$ whose set of unifiers is
not directed, that is, unification for $A$ is not filtering. From [17, Theorem
8.4] we obtain $\mathbf{V}\not\leq\mathbf{V}_{G}$, and thus,
$\operatorname{\mathbf{Eq}}(B_{F})\leq\mathbf{V}$ by the splitting result. ∎

The location of the unification types relative to the splitting
$\langle\operatorname{\mathbf{Eq}}(B_{F}),\mathbf{V}_{G}\rangle$ is shown
below.

$\mathbf{\omega,\;0,\;(\infty?)}$$\mathbf{Eq(B_{F})}$$\mathrm{\mathbf{V}_{CL}}$$\mathbf{1,\;0}$$\mathrm{\mathbf{V}_{G}}$$\mathrm{Eq(2)}$

Therefore, $\operatorname{\mathbf{Eq}}(B_{F})$ is the smallest variety of
closure algebras with finitary but not unitary unification. In a sense the
fork algebra $B_{F}$ plays a role of ‘’a test algebra” for unitary (finitary,
not unitary) unification in varieties of closure algebras: If $\mathbf{V}$ has
unitary (finitary, not unitary) unification, then it does not (does) contain
$B_{F}$. The restriction to varieties with finitary unification is essential:
Dzik et al., [11, 12] presented infinitely many varieties of locally finite
Heyting algebras with unification zero. It can be shown that there are
infinitely many locally finite varieties of Grzegorczyk algebras which have
unification type zero that do not contain $B_{F}$ (and infinitely many that
contain $B_{F}$). This is the topic of ongoing work [9].

##  7 References

##

  * Aiello et al., [2003] Aiello, M., van Benthem, J., and Bezhanishvili, G. (2003).  Reasoning about space: The modal way.  J. Logic Comput., 13(6):889–920.
  * Baader and Snyder, [2001] Baader, F. and Snyder, W. (2001).  Unification theory.  In Robinson, A. and Voronkov, A., editors, Handbook of Automated Reasoning, volume 1, chapter 8, pages 445–532. Elsevier.
  * Balbes and Dwinger, [1974] Balbes, R. and Dwinger, P. (1974).  Distributive lattices.  University of Missouri Press, Columbia, Mo.
  * Blackburn et al., [2001] Blackburn, P., de Rijke, M., and Venema, Y. (2001).  Modal logic, volume 53 of Cambridge Tracts in Theoretical Computer Science.  Cambridge University Press.
  * Blok, [1976] Blok, W. (1976).  Varieties of Interior Algebras.  PhD thesis, University of Amsterdam.
  * Bull and Segerberg, [1984] Bull, R. and Segerberg, K. (1984).  Basic modal logic.  In Gabbay, D. and Guenthner, F., editors, Handbook of Philosophical Logic, pages 1–88. Springer Verlag.
  * Burris, [1995] Burris, S. (1995).  Computers and universal algebra: Some directions.  Algebra Universalis, 34(1):61–71.  MR1344954.
  * Burris and Sankappanavar, [2012] Burris, S. and Sankappanavar, H. P. (2012).  A Course in Universal Algebra - The Millenium Edition, 2012 Update.  Springer Verlag, New York.
  * Düntsch and Dzik, [2023] Düntsch, I. and Dzik, W. (2023).  On the preservation of unification types.  In preparation.
  * Dzik, [2006] Dzik, W. (2006).  Splittings of lattices of theories and unification types.  In Contributions to general algebra, volume 17, pages 71–81. Heyn, Klagenfurt.
  * Dzik et al., [2022] Dzik, W., Kost, S., and Wojtylak, P. (2022).  Finitary unification in locally tabular modal logics characterized.  Ann. Pure Appl. Logic, 173(4):Paper No. 103072.
  * Dzik et al., [2023] Dzik, W., S.Kost, and P.Wojtylak (2023).  Unification types and union splittings in intermediate logics.  Ann. Pure Appl. Logic, to appear.
  * Freese et al., [2011] Freese, R., Kiss, E., and Valeriote, M. (2011).  Universal Algebra Calculator.  Available at http://www.uacalc.org.
  * Frias, [2002] Frias, M. F. (2002).  Fork algebras in algebra, logic and computer science, volume 2 of Advances in Logic.  World Scientific Publishing Co., Inc., River Edge, NJ.
  * Georgiev, [2006] Georgiev, D. (2006).  An implementation of the algorithm SQEMA for computing first-order correspondences of modal formulas.  Master’s thesis, Sofia University, Faculty of Mathematics and ComputerScience.
  * Ghilardi, [1997] Ghilardi, S. (1997).  Unification through projectivity.  J. Logic Comput., 7(6):733–752.
  * Ghilardi and Sacchetti, [2004] Ghilardi, S. and Sacchetti, L. (2004).  Filtering unification and most general unifiers in modal logic.  J. Symbolic Logic, 69(3):879–906.
  * Jónsson and Tarski, [1951] Jónsson, B. and Tarski, A. (1951).  Boolean algebras with operators. I.  Amer. J. Math., 73:891–939.
  * Németi, [1983] Németi, I. (1983).  Surjectiveness of epimorphisms is equivalent to Beth definability property.  Manuscript, Alfréd Rényi Institute of Mathematics, Budapest.
  * Rautenberg, [1980] Rautenberg, W. (1980).  Splitting lattices of logics.  Arch. math. Logik, 20:155–159.