††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Field-Induced Spin Excitations in the Spin-1/2 Triangular-Lattice
Antiferromagnet CsYbSe2

Tao Xie  Neutron Scattering Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37831, USA  Jie Xing  Materials Science and Technology Division, Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  S. E. Nikitin
Paul Scherrer Institut, CH-5232 Villigen, Switzerland  S. Nishimoto
Department of Physics, Technical University Dresden, 01069 Dresden, Germany
Institute for Theoretical Solid State Physics, IFW Dresden, 01069 Dresden,
Germany  M. Brando  Max Planck Institute for Chemical Physics of Solids,
Nöthnitzer Str. 40, D-01187 Dresden, Germany  P. Khanenko  Max Planck
Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, D-01187 Dresden,
Germany  J. Sichelschmidt  Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Str. 40, D-01187 Dresden, Germany  L. D. Sanjeewa
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37831, USA  Athena S. Sefat  Materials Science and Technology
Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  A.
Podlesnyak  Neutron Scattering Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37831, USA

###### Abstract

A layered triangular lattice with spin-1/2 ions is an ideal platform to
explore highly entangled exotic states like quantum spin liquid (QSL). Here,
we report a systematic in-field neutron scattering study on a perfect two-
dimensional triangular-lattice antiferromagnet, CsYbSe2, a member of the large
QSL candidate family rare-earth chalcogenides. The elastic neutron scattering
measured down to 70 mK shows that there is a short-range 120$\degree$ magnetic
order at zero field. In the field-induced ordered states, the spin-spin
correlation lengths along the $c$ axis are relatively short, although the heat
capacity results indicate long-range magnetic orders at 3 T $-$ 5 T. The
inelastic neutron scattering spectra evolve from highly damped continuum-like
excitations at zero field to relatively sharp spin wave modes at the plateau
phase. Our extensive large-cluster density-matrix renormalization group
calculations with a Heisenberg triangular-lattice nearest-neighbor
antiferromagnetic model reproduce the essential features of the experimental
spectra, including continuum-like excitations at zero field, series of sharp
magnons at the plateau phase as well as two-magnon excitations at high energy.
This work presents comprehensive experimental and theoretical overview of the
unconventional field-induced spin dynamics in triangular-lattice Heisenberg
antiferromagnet and thus provides valuable insight into quantum many-body
phenomena.

Geometrically frustrated magnets provide an intriguing playground for
investigation of the novel phenomena in condensed matter physics Diep (1994);
Balents (2010); Rau and Gingras (2019). Strong frustrations may produce large
degeneracy of the ground state and prevent formation of magnetic order in
favor of exotic states such as quantum spin-liquid (QSL) or spin-ice phases
Balents (2010); Rau and Gingras (2019); Broholm  _et al._ (2020).

Two-dimensional (2D) triangular-lattice antiferromagnet (TLAF) with nearest-
neighbor (NN) isotropic antiferromagnetic (AF) coupling is a prototypical
example of frustrated antiferromagnet. In classical regime
($S\rightarrow\infty$) TLAF orders into a long-range 120$\degree$ AF state at
0 K Starykh (2015), while the ground state of TLAF in the quantum limit ($S$ =
1/2) was proposed to be a QSL Anderson (1973); Fazekas and Anderson (1974).
Further theoretical and numerical investigations have shown that $S$ = 1/2
TLAF also shows the 120$\degree$ AF ground state, although the ordered moment
is reduced as compared to the classical case, due to strong quantum
fluctuations Bernu  _et al._ (1992); White and Chernyshev (2007). The quantum
fluctuations also favor collinear up-up-down (UUD) order and stabilize a
magnetization plateau at $M_{s}/3$ ($M_{s}$ is saturation magnetization) for
$S$ = 1/2 TLAF in a finite magnetic field regime Kamiya  _et al._ (2018); Xing
_et al._ (2019); Ranjith  _et al._ (2019a, b); Chubokov and Golosov (1991);
Yamamoto  _et al._ (2014); Schmidt and Thalmeier (2017). Generalization of
this model to the case of anisotropic exchange interactions drastically
enriches the magnetic field-temperature ($B$-$T$) phase diagram and produces a
number of novel quantum phases including the QSL state for certain range of
parameters Li  _et al._ (2016a); Zhu  _et al._ (2018); Maksimov  _et al._
(2019), although the experimental realizations of clean TLAF remain rare.

Recently, Yb-based delafossites $A$Yb$Q_{2}$ ($A$ = Li, Na, K, Rb, Cs, Tl, Ag,
Cu; $Q$ = O, S, Se, Te) [Fig. 1(a)] attracted widespread attention in search
of QSL candidates Liu  _et al._ (2018); Ranjith  _et al._ (2019b, a); Xing
_et al._ (2020); Ding  _et al._ (2019); Ma  _et al._ (2020); Ferreira  _et
al._ (2020); Ranjith  _et al._ ; Iizuka  _et al._ ; Bordelon  _et al._ (2019);
Baenitz  _et al._ (2018); Sichelschmidt  _et al._ ; Dai  _et al._ (2021).
Combination of strong spin-orbit coupling and crystalline electric field
creates an effective spin $S$ = 1/2 at each Yb site at low temperatures due to
Kramers degeneracy of a ground state doublet Baenitz  _et al._ (2018);
Sichelschmidt  _et al._ ; Zhang  _et al._ (2021); Dai  _et al._ (2021). No
long-range order was observed at zero field in $A$Yb$Q_{2}$ by different
measurement techniques, including heat capacity, magnetization, muon
spectroscopy and neutron diffraction Liu  _et al._ (2018); Ranjith  _et al._
(2019b, a); Xing  _et al._ (2019); Ding  _et al._ (2019); Ma  _et al._ (2020);
Ferreira  _et al._ (2020); Ranjith  _et al._ ; Iizuka  _et al._ ; Bordelon
_et al._ (2019). The Yb ions form perfect triangular layers [Fig. 1(a)]
without any intrinsic structural disorder, in contrast to another well-known
TLAF QSL candidate, YbMgGaO4, which shows valuable Mg/Ga site disorders Li
_et al._ (2015a, b, 2016b); Shen  _et al._ (2016); Paddison  _et al._ (2017);
Zhang  _et al._ (2018); Li  _et al._ (2017); Steinhardt  _et al._ (2020); Li
(2019); Shen  _et al._ (2018); Zhu  _et al._ (2017, 2018); Kimchi  _et al._
(2018). Phase diagrams of NaYbSe2 and NaYbO2 Ranjith  _et al._ (2019a);
Bordelon  _et al._ (2019) include a QSL state and magnetic field-induced
ordered states, that makes $A$Yb$Q_{2}$ an ideal platform to study these
different quantum phases in $S$ = 1/2 TLAF.

Low-energy spin dynamics of the Yb-based delafossites have been studied by
single-crystal inelastic neutron scattering (INS) at zero field in CsYbSe2 and
NaYbSe2 Xing  _et al._ (2019); Dai  _et al._ (2021). The signatures of the
gapless excitation continuum, whose intensity concentrates around K point,
$i.e.$, (1/3, 1/3, 0) of Brillouin zone (BZ) and extends up to 2.5 meV, were
found in both materials suggesting a QSL ground state with a spinon Fermi
surface Dai  _et al._ (2021). However, very little is known about spin
dynamics of the field-induced phases in these materials, because all previous
in-field INS measurements were limited by polycrystalline samples Bordelon
_et al._ (2019, 2020). Recent INS study of spin excitations in Ba3CoSb2O9 has
shown that despite the quantum origin of the 1/3 plateau phase in TLAF, its
magnetic excitations can be described using semiclassical nonlinear spin-wave
theory Kamiya  _et al._ (2018).

Thus, even though considerable progresses have been made in theoretical
understanding of TLAF, good experimental realizations remain rare. CsYbSe2 is
one of the best known examples of TLAF without structural disorder SI , with
negligible interlayer coupling and relatively small magnetic energy scale Xing
_et al._ (2019), which allows us to tune the magnetic ground state by moderate
magnetic field. In this Letter, using elastic and inelastic neutron
scattering, heat capacity, and density-matrix renormalization group (DMRG)
methods, we first demonstrate that CsYbSe2 can be described by a nearly
isotropic (Heisenberg) model. We then reveal the pronounced changes in the
magnetic excitation spectrum as it evolves with magnetic field and associate
these changes with the field-driven phase transitions and thus provide
comprehensive overview of spin dynamics in TLAF Heisenberg model.

Figure 1: (a) Crystal structure and the Yb triangular layer of CsYbSe2. (b)
Anisotropy of the ESR $g$ factor at 15 K, the solid line indicates $g(\theta)$
= $\sqrt{g_{ab}^{2}\sin^{2}{\theta}+g_{c}^{2}\cos^{2}{\theta}}$ with $g_{ab}$
= 3.25(0) and $g_{c}$ = 0.3(0). The inset is a typical ESR spectra measured
with $B$ $\parallel$ $ab$, the red solid line in the inset is a Lorentzian
line fitting. (c) The schematic phase diagram extracted from the heat capacity
results. (d)–(e) Temperature dependence of the heat capacity measured at
different magnetic fields with $B$ $\parallel$ $ab$.

We prepared high-quality single crystals of CsYbSe2 using a flux method
according to the previous report Xing  _et al._ (2020). The refinement of
single-crystal X-ray diffraction shows no Cs/Yb site mixing in our crystals SI
. The anisotropy of the $g$ factor was studied by electron spin resonance
(ESR) measurements [Fig. 1(b)], under the same conditions used in other Yb-
based delafossites Sichelschmidt  _et al._ (2019). Heat capacity data with
temperature down to $T=50$ mK and magnetic field up to $B=9$ T were collected
with a semi-adiabatic compensated heat-pulse method in a dilution refrigerator
Kihara  _et al._ (2013). For the INS measurements, about 200 millimeter-sized
single crystals were co-aligned in the $(HHL)$ scattering plane on thin copper
plates to get a mosaic sample with a mass about 0.5 g and mosaicity $\sim$ 2°
SI . The neutron scattering experiments were performed at the time-of-flight
Cold Neutron Chopper Spectrometer (CNCS) Ehlers  _et al._ (2011, 2016) at the
Spallation Neutron Source, Oak Ridge National Laboratory. Measurements were
carried out with an incident neutron energy $E_{\mathrm{i}}$ = 3.32 meV
($\lambda_{\mathrm{i}}$ = 4.96 Å), providing an energy resolution of 0.11 meV
[full width at half maximum (FWHM) at the elastic position]. The magnetic
fields were applied along the vertical direction [$-K$ $K$ 0] using a 5 T
cryomagnet equipped with a dilution refrigerator to cool the sample down to
$T$ = 70 mK. The sample was rotated along the vertical axis in a wide angle
range (210$\degree$) to get a good coverage in both the energy and momentum
space. We used the software packages MantidPlot Arnold  _et al._ (2014) and
Horace Ewings  _et al._ (2016) for data reduction and analysis.

Due to the localized character of 4$f$ states, the effective spin-spin
interactions in CsYbSe2 are mainly limited by the NN exchange. The general NN
anisotropic-exchange Hamiltonian constrained by the symmetries of the
triangular lattice can be written as Li  _et al._ (2015b, 2016a, 2018); Zhu
_et al._ (2018); Maksimov  _et al._ (2019)

$\displaystyle\mathcal{H}=J\sum_{\langle
i,j\rangle}({S}^{x}_{i}{S}^{x}_{j}+{S}^{y}_{i}{S}^{y}_{j}+\Delta{S}^{z}_{i}{S}^{z}_{j})-\mu_{B}g_{ab}B\sum_{i}{S}^{y}_{i}$
$\displaystyle+\sum_{\langle
i,j\rangle}2J_{\pm\pm}[(S^{x}_{i}S^{x}_{j}-S^{y}_{i}S^{y}_{j})\tilde{c}_{\alpha}-(S^{x}_{i}S^{y}_{j}+S^{y}_{i}S^{x}_{j})\tilde{s}_{\alpha}]$
$\displaystyle+\sum_{\langle
i,j\rangle}J_{z\pm}[(S^{y}_{i}S^{z}_{j}+S^{z}_{i}S^{y}_{j})\tilde{c}_{\alpha}-(S^{x}_{i}S^{z}_{j}+S^{z}_{i}S^{x}_{j})\tilde{s}_{\alpha}],$
(1)

where $\langle i,j\rangle$ denotes NN sites, $0\leq\Delta\leq 1$, $\mu_{B}$ is
the Bohr magneton, $g_{ab}$ is the in-plane $g$-tensor component, $B$ is the
strength of the magnetic induction, which is parallel to the in-plane $\bf{y}$
direction [Fig. 1(a)]. $\tilde{c}_{\alpha}=\cos{\tilde{\varphi}_{\alpha}}$ and
$\tilde{s}_{\alpha}=\sin{\tilde{\varphi}_{\alpha}}$, in which bond angles
$\tilde{\varphi}_{\alpha}$ are angles between the primitive vectors of the
lattice $\bm{\delta}_{\alpha}$ and the $x$ axis [Fig. 1(a)],
$\tilde{\varphi}_{\alpha}=\\{0,2\pi/3,-2\pi/3\\}$. The exchange interaction
parameters follow, $J_{zy}=J_{yz}=J_{z\pm}$, $J_{zz}=\Delta J$, with
$J=(J_{xx}+J_{yy})/2$ and $J_{\pm\pm}=(J_{xx}-J_{yy})/4$ Maksimov  _et al._
(2019).

Hamiltonian (Field-Induced Spin Excitations in the Spin-1/2 Triangular-Lattice
Antiferromagnet CsYbSe2) has been extensively explored to establish the phase
diagram of the general NN TLAF model Li  _et al._ (2016a); Luo  _et al._
(2017); Zhu  _et al._ (2017, 2018); Iaconis  _et al._ (2018); Maksimov  _et
al._ (2019). It was found that the phase diagram is dominated by magnetically
ordered single-$\mathbf{Q}$ phases, which are ferromagnetic, 120∘
N$\acute{e}e$l, dual 120∘, and two different stripe states Maksimov  _et al._
(2019). For CsYbSe2, from the fitting of the experimental magnetization curve
and the width of the 1/3 magnetization plateau with grand canonical DMRG
method, we found that the XXZ anisotropy is small, $i.e.$, $\Delta\approx 1$
(see details in SI ). The best fitting parameters are $J$ = 0.544 meV,
$g_{ab}$ = 4.148. To examine the dynamical properties of the system (Field-
Induced Spin Excitations in the Spin-1/2 Triangular-Lattice Antiferromagnet
CsYbSe2), dynamical DMRG calculations Jeckelmann (2002); Nishimoto and Arikawa
(2007) on 61-site open clusters under open boundary conditions were performed
(see details in SI ). Having in mind that (i) we did not observe the
nonreciprocal spin-wave spectra which is caused by the presence of anisotropic
bond-dependent interaction $J_{z\pm}$ Maksimov  _et al._ (2019) (see details
in SI ), and (ii) $J_{z\pm}$ and $J_{\pm\pm}$ are expected to be small
compared to $J$ Ranjith  _et al._ (2019a), we considered the bond-independent
XXZ part of Hamiltonian (Field-Induced Spin Excitations in the Spin-1/2
Triangular-Lattice Antiferromagnet CsYbSe2) only, neglecting the bond-
dependent anisotropic terms.

Figure 2:  Elastic neutron scattering signals ($E$ = 0 $\pm$ 0.05) in $(HHL)$
plane at 70 mK. (a) 1D constant energy cuts along [$H$ $H$ 1]. (b) 1D constant
energy cuts along [1/3 1/3 $L$]. The solid curves are fitting results of the
Voigt function with linear backgrounds. The data points and fitting curves
have been shifted vertically for clarity. (c) 2D zero-energy slices in $(HHL)$
plane for 0 T (left-half part) and 4 T (right-half part). In panels (a)–(c),
the integrated range along [$-K$ $K$ 0] direction is $K$ = [$-$0.05, 0.05].
(d) The in-plane and out-of-plane correlation length obtained from the Fourier
transform of the magnetic Bragg peak width at $Q=(1/3,1/3,1)$.

In order to reveal which phases are stabilized at base temperature in
different magnetic fields, we first analyze the elastic neutron scattering
signal (obtained as neutron energy transfer
$E=\hbar\omega=E_{\mathrm{i}}-E_{\mathrm{f}}=0\pm 0.05$ meV) in complement to
the heat capacity data. The heat capacity as a function of temperature at
different in-plane magnetic fields are presented in Figs. 1(c) and 1(d). For
the data shown in Fig. 1(c), sharp $\lambda$-shape peaks that correspond to
phase transitions of long-range orders can be identified in 3 T – 5 T. While
for the curves measured at magnetic fields $B\leq 2$ T, and $B\geq 6$ T, only
broad peaks/humps that indicate short-range correlations/orders can be
observed [Fig. 1(d)].

Elastic neutron scattering at 70 mK shows that weak magnetic order has formed
at 0 T [Figs. 2(a)–2(c)]. In the two-dimensional (2D) constant energy slice at
0 T [left half in Fig. 2(c)], we can see weak intensity appears at $Q$ = (1/3,
1/3, $L$) ($L$ = odd integers), which correspond to a propagation vector
$\bf{q}$ = (1/3, 1/3, 1). Since this weak order appears at zero field, we
attribute it to the well-known 120∘ ordered state for the Heisenberg $S$ = 1/2
TLAF. At 4 T, we can see clearly rod of magnetic intensity elongated along the
$[1/3~{}1/3~{}L]$ direction and peaked at $Q$ = (1/3, 1/3, 1) and equivalent
positions [right half of Fig. 2(c)]. To characterize the peaks quantitatively
we made the one-dimensional (1D) cuts along $[H~{}H~{}1]$ and
$[1/3~{}1/3~{}L]$ at $Q$ = (1/3, 1/3, 1) [Figs. 2(a) and 2(b)]. At 0 T, the
weak peak is broad along both directions and application of 1 T field
suppresses it further. Above 1 T, the magnetic peak evolves to be sharper and
stronger with the increasing magnetic field and reaches a maximal intensity at
4 T. Further field increase up to 5 T causes minor decrease of the intensity.
The data points shown in Figs. 2(a) and 2(b) can be well fitted by Voigt
function with a linear background SI . In order to make the spin-spin
correlation more intuitive, we convert the peak width at $Q$ = (1/3, 1/3, 1)
to the spin-spin correlation length ($\xi$) in real space taking into account
the instrumental resolution, and the results are shown in Fig. 2(d) SI . For
all the measured fields, $\xi_{c}$ $\textless$ 36 Å ($\sim$ 2.2$c$, $c$ is
lattice constant), which indicates a short-range spin-spin correlation along
the $c$ axis. The spin-spin correlation in $ab$ plane also starts from short-
range at 0 T ($\xi_{ab}\approx$ 60 Å), reaches a minimum at 1 T
($\xi_{ab}\approx$ 34 Å) and then evolves to be long-range above 3 T with
$\xi_{ab}$ $\approx$ 245 Å ($\sim$ 60$a$, $a$ is lattice constant). The
evolution of $\xi_{ab}$ is consistent with the heat capacity results in Figs.
1(c) and 1(d), where sharp $\lambda$-shape peaks can only be observed in the
long-range ordered states at 3 T – 5 T, and the broad humps at 0 T – 2 T are
related to formation of the short-range magnetic orders.

Figure 3:  Spin excitation spectra along $[H~{}H]$ direction at different
magnetic fields at 70 mK. (a) The schematic of the Brillouin zones in the
$(H~{}K~{}0)$ plane. The high symmetry points and high symmetry directions are
indicated by the dots and arrows, respectively. (b)–(f) Experimental (left-
half panels) and calculated (right-half panels) dynamical spin structure
factors $S(\mathbf{Q},\omega)$ at different magnetic fields. All the
experimental data shown here have been symmetrized according to the crystal
symmetry. The integrated range along [$-K$ $K$] direction is $K$ = [$-$0.1,
0.1]. The $y$ components of the spin quantum numbers $S_{y}$ = 9/2, 23/2,
25/2, and 27/2 are used for indicating the strength of the magnetic fields in
dynamical DMRG calculations SI .

According to our above analyses and the results in Ref. Ranjith  _et al._
(2019a), we propose the following scenario for field-induced evolution of the
magnetic structures in CsYbSe2. At base temperature, the system starts from a
short-range 120$\degree$ magnetic order at 0 T, then gradually evolves to the
oblique 120$\degree$ ($Y$-coplanar) phase near 2 T < $B$ < 3 T, and the
collinear UUD order for $B$ $\geq$ 3 T. For higher field strength like 5 T and
6 T, the system is in a canted 2:1 coplanar state Ranjith  _et al._ (2019a);
Garlea  _et al._ (2019). Combining the neutron diffraction results with the
heat capacity data, we depict a schematic $B$-$T$ phase diagram [Fig. 1(b)].
The solid and open circles in the phase diagram correspond to the temperatures
of the sharp peaks and broad humps in the heat capacity data shown in Figs.
1(c) and 1(d), respectively.

We now turn to the field dependence of the dynamical properties of the TLAF
CsYbSe2. Figure 3 summarizes the experimental spin excitation spectra (left-
half panels) along the high symmetry direction [$H$ $H$] [the horizontal arrow
in Fig. 3(a)] at different magnetic fields in agreement with the corresponding
DMRG calculations (right-half panels). Since the spin excitation has no
intensity modulation and energy dispersion along [0 0 $L$] direction SI , we
integrated a wide $L$ range (1.5 $\leq$ $|\mathrm{L}|$ $\leq$ 5.5) to increase
the statistics. The background of the spectra has been subtracted, and the
details of the background subtraction can be found in supplemental material SI
.

Similar to the previous reports Xing  _et al._ (2019); Dai  _et al._ (2021),
the INS spectrum measured at 0 T shows diffusive continuum-like spin
excitations accumulated around K point of the Brillouin zone [Fig. 3(b)].
Except for the diffusive continuum-like excitation, the spectrum contains a
branch of weak and diffusive spin excitation stems from the $\Gamma$ point. At
2 T, the spectrum opens a clear gap around the K point below $E$ $\sim$ 0.4
meV [Fig. 3(c)]. Below the gap energy, there is another low-energy mode that
merges with the incoherent background. Further increasing the field to 3 T,
the most impressive observation is that a strong and sharp spin-wave mode
appears in the energy range 0.55 $\textless$ $E$ $\textless$ 0.8 meV (mode
III) [Fig. 3(d)]. Additionally, three other excitation branches can be
identified. The lowest branch (mode I) with band top at $\sim$ 0.4 meV is very
weak and visible only in the momentum range $-$0.6 $\textless$ $H$ $\textless$
$-$0.4. The branch (mode II) between the mode I and the mode III concentrates
its intensity around the K point and touches the mode III around 0.5 meV. The
highest diffusive branch (mode IV) locates in the energy range 0.8 $\textless$
$E$ $\textless$ 1.5 meV. At 4 T, the mode I become very strong and the energy
top is lifted to $\sim$ 0.7 meV, while the mode II disappears, and the mode
III is lifted to 0.75 $\textless$ $E$ $\textless$ 1 meV. The mode IV keeps
almost unchanged both in intensity and energy range. At the highest measured
magnetic field, 5 T, the mode I becomes weaker than at 4 T. While, in the
energy range 0.8 $\textless$ $E$ $\textless$ 1.2 meV, the mode III evolves to
be a bow-tie-like continuum with intensity maximum located at $\Gamma$ and K
point. The weakest mode IV seems to be still existent, but becomes weaker and
merges with the bow-tie-like continuum.

Previously, the spin dynamics in TLAF were described by different models,
among which linear and non-linear spin wave theory (SWT) are the most
conventional tools Ma  _et al._ (2016); Ito  _et al._ (2017); Mourigal  _et
al._ (2013); Zheng  _et al._ (2006); Macdougal  _et al._ (2020). For instance,
non-linear SWT was successfully applied to simulate the dispersion in TLAF
Ba3CoSb2O9 within the field-induced UUD phase Kamiya  _et al._ (2018).
However, this model fails to reproduce the broadening magnon linewidth due to
strong magnon-magnon interactions in the 120$\degree$ phase at zero field Ma
_et al._ (2016); Ito  _et al._ (2017). Therefore, in this work we used a
conceptually different approach and performed large-cluster DMRG method to
calculate the spin dynamics explicitly at different fields (see supplemental
material SI  for details of DMRG calculations). The used exchange coupling
parameter in the dynamical DMRG calculations is $J$ = 0.48 meV. The
anisotropic components of $g$ factor ($g_{ab}$ = 3.25, $g_{c}$ = 0.3)
extracted from the ESR measurements [Fig. 1(b)] have been used in the
calculations. As we show below, this simple model treated by dynamical DMRG
provides surprisingly good qualitative account of the observed INS spectra.

The calculated dynamical spin structure factors, $S(\mathbf{Q},\omega)$, are
shown in Figs. 3(b)–3(f) along with the experimental spectra. For the low-
field regime below the UUD phase as shown in Fig. 3(b) and 3(c), our model
well captures the overall behavior, including the strong, gapless, damped
excitations at zero field and the two diffusive excitation modes with an
energy gap between them at 2 T. At $B=3$ T and 4 T, upon entering the
collinear UUD phase, the calculated $S(\mathbf{Q},\omega)$ not only reproduce
the three stronger low-energy branches (modes I–III), but also gives
reasonable description of the high-energy weak branch (mode IV) [Figs.
3(c)–3(d)]. It is worth noting that the calculated intensity of modes I–III
mostly comes from the transverse (perpendicular to the direction of the
magnetic field, $i.e.$, $\mathbf{y}$) structure factors
$S^{xx}(\mathbf{Q},\omega)$ and $S^{zz}(\mathbf{Q},\omega)$, while the
intensity of mode IV is mostly contributed by the longitudinal structure
factor $S^{yy}(\mathbf{Q},\omega)$ (see supplemental information SI  for
different components of the calculated dynamical structure factors). This
means that the modes I–III correspond to the single-magnon excitations, the
weak and diffusive mode IV corresponds to the two-magnon continuum Kamiya  _et
al._ (2018); Chernyshev and Zhitomirsky (2009). Above the UUD phase at $B=5$
T, the low-energy mode, and the high-energy bow-tie-like continuum with
resonance-like maximal intensity at $\Gamma$ and K points of the BZ are
qualitatively reproduced by our DMRG calculation again [Fig. 3(f)]. It should
be noticed that the energy scale of the calculated spectra at some non-zero
fields does not precisely match the experimental ones and we will discuss
possible origins of the minor disagreements below.

The DMRG simulations of the Heisenberg TLAF model reproduce the major spectral
features of the measured excitations: (i) broad overdamped excitations at low-
field 120∘ state [Fig. 3(b)]; (ii) field-induced spin gap and the additional
mode stems from $\Gamma$ point [Fig. 3(c)]; (iii) series of magnon modes and
drastic increase of lifetime of magnons within the plateau phase [Figs. 3(d)
and 3(e)]; (iv) decay of magnon mode and resonance-like feature at K and
$\Gamma$ points above the plateau phase [Fig. 3(f)]; (v) multimagnon
excitations at high energy [Figs. 3(d)–3(f)]. The consistency between
experimental and numerical data indicates that the spin dynamics in CsYbSe2 is
primarily dictated by the isotropic exchange interaction. Therefore, our
results provide comprehensive overview of field-induced spin dynamics in this
perfect $S$ = 1/2 TLAF. The quantitative discrepancy between experimental data
and the calculated spectra shown in Fig. 3 can be caused by the
oversimplification of our model, which does not go beyond simple Heisinberg
interaction as well as by finite-size effects in calculations. Another
possible reason is that the strength of used magnetic fields in the
calculations of $S(\mathbf{Q},\omega)$ do not match all the experimental
fields perfectly SI .

We further note that the zero-field spectrum shown in Fig. 3(b) looks very
similar to the continuum-like excitation observed in isostructural NaYbSe2,
which is claimed to be a QSL with a spinon Fermi surface Dai  _et al._ (2021).
As we have mentioned above, CsYbSe2 is a perfect TLAF without disorder, the
continuum-like excitation in Fig. 3(b) cannot be a disorder-induced mimicry of
QSL that was proposed in Ref. Zhu  _et al._ (2017). Previous theoretical
studies on TLAF have suggested that the QSL phase occurs at the border between
120$\degree$ and stripe phases, and the maximal extent of the QSL phase is
achieved at the isotropic limit of the XXZ term Zhu  _et al._ (2017, 2018). As
the short-range 120$\degree$ magnetic order has been confirmed at zero field
by elastic neutron scattering and the INS spectra has been reproduced by the
DMRG calculations with an isotropic Heisenberg TLAF model, CsYbSe2 may be
located in a critical region between the QSL phase and 120$\degree$ phase at
zero field. In this region the system is so unstable that even a weak
perturbation of quantum fluctuations will push the system towards a QSL or
magnetically ordered ground state.

In summary, we observed a short-range 120$\degree$ magnetic order at zero
field in the $S$ = 1/2 TLAF CsYbSe2. A schematic $B$-$T$ phase diagram was
depicted on the basis of heat capacity and neutron diffraction results. The
coexistence of short-range 120$\degree$ magnetic order and highly damped
continuum-like excitations at zero field suggests that CsYbSe2 may be in a
critical region between the QSL phase and 120$\degree$ phase. We report the
first in-field single-crystal INS study on the family of Yb-based
dichalcogenide delafossites $A$Yb$Q_{2}$. The INS spectra evolve from the
zero-field continuum-like excitations to relatively sharp spin wave modes in
field-induced phases. The spin excitation spectra at different fields can be
well described by an isotropic Heisenberg TLAF model with the dynamical DMRG
method. This work provides a nearly isotropic NN Hamiltonian that is
appropriate for the different quantum phases in an ideal $S$ = 1/2 TLAF
compound, and sheds light on the understanding of the physics of TLAF
compounds.

## Acknowledgments

We thank Colin McMillen for the help with the single-crystal X-ray diffraction
refinement, U. Nitzsche for technical assistance, and B. Schmidt for the
fruitful discussion. The research at the Oak Ridge National Laboratory (ORNL)
is supported by the U.S. Department of Energy (DOE), Office of Science, Basic
Energy Sciences (BES), Materials Sciences and Engineering Division. This
research used resources at Spallation Neutron Source, a DOE Office of Science
User Facility operated by ORNL. X-ray Laue alignment was conducted at the
Center for Nanophase Materials Sciences (CNMS) (CNMS2019-R18) at ORNL, which
is a DOE Office of Science User Facility. S.E.N. acknowledges funding from the
European Unions Horizon 2020 research and innovation program under the Marie
Skłodowska-Curie grant agreement No 884104. S.N. acknowledges support by the
Deutsche Forschungsgemeinschaft through SFB 1143 project no. A05.

This paper has been authored by UT-Battelle, LLC under Contract No. DE-
AC05-00OR22725 with the U.S. Department of Energy. The United States
Government retains and the publisher, by accepting the article for
publication, acknowledges that the United States Government retains a non-
exclusive, paid-up, irrevocable, world-wide license to publish or reproduce
the published form of this manuscript, or allow others to do so, for United
States Government purposes. The Department of Energy will provide public
access to these results of federally sponsored research in accordance with the
DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

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11footnotetext: Support of the research by the Austrian Science Fund (FWF),
project I 4579-N, and the Czech Science Foundation (GAČR), project 20-09869L,
entitled “The many facets of orthomodularity”, as well as by ÖAD, project CZ
02/2019, entitled “Function algebras and ordered structures related to logic
and data fusion”, and, concerning the first author, by IGA, project PřF 2021
030, is gratefully acknowledged.22footnotetext: Corresponding author.

# Join-semilattices whose principal filters are pseudocomplemented lattices

Ivan Chajda and Helmut Länger2

###### Abstract

This paper deals with join-semilattices whose sections, i.e. principal
filters, are pseudocomplemented lattices. The pseudocomplement of $a\vee b$ in
the section $[b,1]$ is denoted by $a\rightarrow b$ and can be considered as
the connective implication in a certain kind of intuitionistic logic. Contrary
to the case of Brouwerian semilattices, sections need not be distributive
lattices. This essentially allows possible applications in non-classical
logics. We present a connection of the semilattices mentioned in the beginning
with so-called non-classical implication semilattices which can be converted
into I-algebras having everywhere defined operations. Moreover, we relate our
structures to sectionally and relatively residuated semilattices which means
that our logical structures are closely connected with substructural logics.
We show that I-algebras form a congruence distributive, $3$-permutable and
weakly regular variety.

AMS Subject Classification: 06A12, 06D15, 03G25, 03B47

Keywords: Join-semilattice, pseudocomplemented lattice, intuitionistic
implication,
non-classical implication semilattice, sectionally residuated join-
semilattice, relatively residuated join-semilattice, I-algebra, variety of
I-algebras, congruence distributive, 3-permutable, weakly regular, R-algebra.

##  1 Introduction

The concept of implication algebra was introduced by J. C. Abbott ([1]) under
the name semi-boolean algebra. It is in fact an axiomatization of the
connective implication in classical logic, which was formalized by means of a
Boolean algebra already by G. Boole. As shown by Abbott, this algebra is in
fact a join-semilattice with top element $1$ where every principal filter, the
so-called section $[x,1]$, is a Boolean algebra. Hence, the operation $\vee$
is everywhere defined and it can be expressed by means of implication $x\cdot
y$ as $x\vee y=(x\cdot y)\cdot y$. On the other hand, the operation $\wedge$
is only partial and $x\wedge y$ is defined if and only if the elements $x,y$
have a common lower bound $p$; in such a case $x\wedge y=\big{(}x\cdot(y\cdot
p)\big{)}\cdot p$ and the complement of $x$ in $[p,1]$ is $x^{p}=x\cdot p$.

Later on, Abbott formalized in a similar way also the connective implication
in the logic of quantum mechanics based on an orthomodular lattice, see [2].
In an analogous way, implication was axiomatized in ortholattices and in
orthomodular lattices without the compatibility condition by the authors and
R. Halaš in [9] and [10], see also [5] and [7] for similar cases.

Concerning implication semilattices, i.e. join-semilattices where every
section is a Boolean algebra, and concerning the structure and properties of
such algebras, the reader is referred to [15], [16] and [17].

It is a natural question whether the connective implication in intuitionistic
logic can be axiomatized in a similar way. It is known that the so-called
Brouwerian semilattices can serve for this sake, see e.g. [13]. In this case
we obtain a join-semilattice with top element $1$ whose sections are
relatively pseudocomplemented posets or lattices, see [3] for these concepts.
However, relatively pseudocomplemented lattices are distributive (see e.g.
[3]), this fact restricting their application in non-classical logics.

An attempt to extend the concept of relative pseudocomplementation to non-
distributive lattices was made by the first author in [6]. This kind of
pseudocomplementation was called sectional pseudocomplementation and it was
extended also to posets with top element $1$ by the authors and J. Paseka in
[12].

Hence, we can extend our investigation of intuitionistic implication also to
semilattices whose sections need not be distributive lattices. We show that in
this case, these join-semilattices can be converted into the so-called
sectionally residuated join-semilattices which are in one-to-one
correspondence with relatively residuated join-semilattices introduced by the
authors and J. Kühr for lattices in [11].

Since in our structures the lattice operation $\wedge$ is only partial, the
algebras under consideration are determined by so-called partial identities
and hence they form the so-called partial varieties. To avoid drawbacks with
partial operations, we introduce a new ternary everywhere defined operation
and we show that our non-classical implication algebra can be converted in an
algebra with everywhere defined operations characterized by identities only,
and hence the corresponding class forms a variety. We show that this variety
has nice congruence properties.

We believe that our approach brings a new insight in this topic and enables to
understand the structure of investigated objects.

##  2 Basic concepts

A pseudocomplemented lattice (see e.g. [3] and [4]) is an algebra
$(L,\vee,\wedge,{}^{*},0)$ of type $(2,2,1,0)$ such that $(L,\vee,\wedge,0)$
is a lattice with bottom element $0$ and for every $x\in L$, $x^{*}$ is the
greatest element $y\in L$ satisfying $x\wedge y=0$, the so-called
pseudocomplement of $x$. It is easy to see that $x\leq y$ implies $y^{*}\leq
x^{*}$ and that $x\leq x^{**}$.

Let $(S,\vee,1)$ be a join-semilattice with top element $1$ whose principal
filters are pseudocomplemented lattices. This means that for every $x\in S$
there exists a binary operation $\wedge_{x}$ and a unary operation x on
$[x,1]$ such that $([x,1],\vee,\wedge_{x},{}^{x})$ is a pseudocomplemented
lattice with bottom element $x$. It is natural to assume the following
compatibility condition:

$x\leq y\text{ and }z,u\in[y,1]\Rightarrow z\wedge_{x}u=z\wedge_{y}u.$

We then also have

$z,u\in[x,1]\cap[y,1]\Rightarrow z\wedge_{x}u=z\wedge_{y}u$

since $z,u\in[x,1]\cap[y,1]$ implies $z,u\in[x,1]\cap[y,1]\cap[x\vee y,1]$ and
hence

$z\wedge_{x}u=z\wedge_{x\vee y}u=z\wedge_{y}u.$

In the following $z\wedge u$ denotes the infimum of $z$ and $u$ provided it
exists. In general, the binary operation $\wedge$ is only partial. But now we
can prove

$z,u\in[x,1]\Rightarrow z\wedge_{x}u=z\wedge u.$

This can be seen as follows: Because of $z,u\in[x,1]$ we have
$z\wedge_{x}u\leq z,u$. If $y\in S$ and $y\leq z,u$ then $z,u\in[y,1]$ and
hence $y\leq z\wedge_{y}u=z\wedge_{x}u$. This shows $z\wedge_{x}u=z\wedge u$.
Hence $z\wedge u$ exists if and only if $z$ and $u$ have a common lower bound.
This motivates the following definition.

##  3 Join-semilattices with pseudocomplemented sec-
tions

We start with the following fundamental definition.

######  Definition 3.1.

A join-semilattice whose sections are pseudocomplemented lattices is an
ordered quintuple $(S,\vee,\wedge,(^{x};x\in S),1)$ where $(S,\vee,1)$ is a
join-semilattice with top element $1$, $x\wedge y$ is defined if and only if
$x$ and $y$ have a common lower bound and for every $x\in S$,
$([x,1],\vee,\wedge,{}^{x})$ is a pseudocomplemented lattice.

Here and in the following the word “section” means the same as “principal
filter”.

In order to avoid problems with such a number of unary operations
(pseudocomplementations in sections), we introduce a new binary operation
$\rightarrow$ satisfying four simple axioms. This new structure would
represent an implication reduct of a certain kind of non-classical
(intuitionistic) logic. Because the name implication semilattice was already
used by W. C. Nemitz (see e.g. [15], [16] and [17]) for a semilattice whose
sections are Boolean algebras (which formalizes the implication reduct of
classical logic), we use a different name for our structure. Hence, we define.

######  Definition 3.2.

An non-classical implication semilattice is an ordered quintuple
$(S,\vee,\wedge,$ $\rightarrow,1)$ where $(S,\vee,1)$ is a join-semilattice
with top element $1$, $x\wedge y$ exists if and only if $x$ and $y$ have a
common lower bound and $\rightarrow$ is a binary operation on $S$ satisfying
(1) – (4):

  1. (1)

$y\leq x\rightarrow y$,

  2. (2)

$(x\vee y)\wedge(x\rightarrow y)\approx y$,

  3. (3)

$(x\vee y)\rightarrow y\approx x\rightarrow y$,

  4. (4)

$y\leq(x\vee z)\rightarrow\big{(}(x\vee z)\wedge(y\vee z)\big{)}$.

######  Remark 3.3.

Conditions (1) and (3) are identities whereas (2) and (4) can be considered as
identities where the operation $\wedge$ is only partial. Hence the class of
non-classical implication semilattices forms a partial variety.

Observe that contrary to our case the sections of a relatively residuated
join-semilattice need not be lattices and hence the meet of two elements which
have a common lower bound need not exist.

For every join-semilattice $\mathbf{S}=(S,\vee,\wedge,(^{x};x\in S),1)$ whose
sections are pseudocomplemented lattices put
$\mathbb{I}(\mathbf{S}):=(S,\vee,\wedge,\rightarrow,1)$ where $\rightarrow$
denotes the binary operation on $S$ defined by $x\rightarrow y:=(x\vee y)^{y}$
for all $x,y\in S$.

We can relate this concept with that one defined above.

######  Theorem 3.4.

Let $\mathbf{S}=(S,\vee,\wedge,(^{x};x\in S),1)$ be a join-semilattice whose
sections are pseudocomplemented lattices. Then
$\mathbb{I}(\mathbf{S})=(S,\vee,\wedge,\rightarrow,1)$ is a non-classical
implication semilattice satisfying (5) – (9):

  1. (5)

$x\leq y\Leftrightarrow x\rightarrow y=1$,

  2. (6)

$x\leq y\Rightarrow y\rightarrow z\leq x\rightarrow z$,

  3. (7)

$x\leq(x\rightarrow y)\rightarrow y$,

  4. (8)

$\big{(}(x\rightarrow y)\rightarrow y\big{)}\rightarrow y\approx x\rightarrow
y$,

  5. (9)

$1\rightarrow x\approx x$.

###### Proof.

Let $a,b,c\in S$.

  1. (1)

$b\leq(a\vee b)^{b}=a\rightarrow b$

  2. (2)

$(a\vee b)\wedge(a\rightarrow b)=(a\vee b)\wedge(a\vee b)^{b}=b$

  3. (3)

$(a\vee b)\rightarrow b=\big{(}(a\vee b)\vee b\big{)}^{b}=(a\vee
b)^{b}=a\rightarrow b$

  4. (4)

Since $b\vee c\in[(a\vee c)\wedge(b\vee c),1]$ and $(a\vee c)\wedge(b\vee
c)=(a\vee c)\wedge(b\vee c)$ we have

$b\vee c\leq(a\vee c)^{(a\vee c)\wedge(b\vee c)}$

and hence

$\displaystyle b$ $\displaystyle\leq b\vee c\leq(a\vee c)^{(a\vee
c)\wedge(b\vee c)}=\Big{(}a\vee c\vee\big{(}(a\vee c)\wedge(b\vee
c)\big{)}\Big{)}^{(a\vee c)\wedge(b\vee c)}=$ $\displaystyle=(a\vee
c)\rightarrow\big{(}(a\vee c)\wedge(b\vee c)\big{)}$

  5. (5)

If $a\leq b$ then $a\rightarrow b=(a\vee b)^{b}=b^{b}=1$. If, conversely,
$a\rightarrow b=1$ then $a\leq a\vee b=(a\vee b)\wedge 1=(a\vee
b)\wedge(a\rightarrow b)=(a\vee b)\wedge(a\vee b)^{b}=b$.

  6. (6)

If $a\leq b$ then $b\rightarrow c=(b\vee c)^{c}\leq(a\vee c)^{c}=a\rightarrow
c$.

  7. (7)

$a\leq a\vee b\leq(a\vee b)^{bb}=\big{(}(a\vee b)^{b}\vee
b\big{)}^{b}=(a\rightarrow b)\rightarrow b$

  8. (8)

By (7) and (6) we have

$a\rightarrow b\leq\big{(}(a\rightarrow b)\rightarrow b\big{)}\rightarrow
b\leq a\rightarrow b,$

i.e.

$\big{(}(a\rightarrow b)\rightarrow b\big{)}\rightarrow b=a\rightarrow b.$

  9. (9)

$1\rightarrow a=(1\vee a)^{a}=1^{a}=a$

∎

The following example shows that in a join-semilattice whose sections are
pseudocomplemented lattices these lattices need neither be distributive nor
modular.

######  Example 3.5.

The join-semilattice depictured in Figure 2

$0$$b$$a$$c$$d$$1$Fig. 2

is a join-semilattice whose sections are pseudocomplemented lattices and the
operation tables for $\rightarrow$ and $\wedge$ look as follows:

$\begin{array}[]{c|cccccc}\rightarrow&0&a&b&c&d&1\\\ \hline\cr
0&1&1&1&1&1&1\\\ a&b&1&b&1&d&1\\\ b&c&a&1&c&d&1\\\ c&b&a&b&1&d&1\\\
d&0&a&b&c&1&1\\\
1&0&a&b&c&d&1\end{array}\quad\quad\quad\begin{array}[]{c|cccccc}\wedge&0&a&b&c&d&1\\\
\hline\cr 0&0&0&0&0&-&0\\\ a&0&a&0&a&-&a\\\ b&0&0&b&0&-&b\\\ c&0&a&0&c&-&c\\\
d&-&-&-&-&d&d\\\ 1&0&a&b&c&d&1\end{array}$

In what follows we show the converse way from a non-classical implication
semilattice to a join-semilattice whose sections are pseudocomplemented
lattices. For this sake, let us introduce the following concept.

For every non-classical implication semilattice
$\mathbf{I}=(I,\vee,\wedge,\rightarrow,1)$ put

$\mathbb{S}(\mathbf{I}):=(I,\vee,\wedge,(^{x};x\in I),1)$

where for every $x\in I$, x denotes the unary operation on $[x,1]$ defined by
$y^{x}:=y\rightarrow x$ for all $y\in[x,1]$.

######  Theorem 3.6.

Let $\mathbf{I}=(I,\vee,\wedge,\rightarrow,1)$ be a non-classical implication
semilattice. Then $\mathbb{S}(\mathbf{I})$ is a join-semilattice whose
sections are pseudocomplemented lattices.

###### Proof.

Let $\mathbb{S}(\mathbf{I})=(I,\vee,\wedge,(^{x};x\in I),1)$ and $a,b,c\in I$
with $c\leq a,b$. If $a\wedge b=c$ then according to (4) we obtain

$b\leq(a\vee c)\rightarrow\big{(}(a\vee c)\wedge(b\vee
c)\big{)}=a\rightarrow(a\wedge b)=a\rightarrow c=a^{c}.$

If, conversely, if $b\leq a^{c}$ then

$c\leq a\wedge b\leq a\wedge a^{c}\leq(a\vee c)\wedge(a\rightarrow c)=c$

according to (2) and hence $a\wedge b=c$. Hence, for each $a\in[c,1]$ the
element $a\rightarrow c$ is the pseudocomplement of $a$ in this interval. ∎

We justify our conversions by the following result.

######  Theorem 3.7.

The correspondence described in Theorems 3.4 and 3.6 is one-to-one.

###### Proof.

If $\mathbf{S}=\big{(}S,\vee,\wedge,(^{x};x\in S),1\big{)}$ is a join-
semilattice whose sections are pseudocomplemented lattices and

$\displaystyle\mathbb{I}(\mathbf{S})$
$\displaystyle=(S,\vee,\wedge,\rightarrow,1),$
$\displaystyle\mathbb{S}\big{(}\mathbb{I}(\mathbf{S})\big{)}$
$\displaystyle=\big{(}S,\vee,\wedge,(_{x};x\in S),1\big{)}$

then $y_{x}=y\rightarrow x=(y\vee x)^{x}=y^{x}$ for all $x\in S$ and
$y\in[x,1]$ and hence
$\mathbb{S}\big{(}\mathbb{I}(\mathbf{S})\big{)}=\mathbf{S}$. If, conversely,
$\mathbf{I}=(I,\vee,\wedge,\rightarrow,1)$ is a non-classical implication
semilattice and

$\displaystyle\mathbb{S}(\mathbf{I})$
$\displaystyle=\big{(}I,\vee,\wedge,(^{x};x\in S),1\big{)},$
$\displaystyle\mathbb{I}\big{(}\mathbb{S}(\mathbf{I})\big{)}$
$\displaystyle=(I,\vee,\wedge,\Rightarrow,1)$

then $x\Rightarrow y\approx(x\vee y)^{y}\approx(x\vee y)\rightarrow y\approx
x\rightarrow y$ according to (3) and hence
$\mathbb{I}\big{(}\mathbb{S}(\mathbf{I})\big{)}=\mathbf{I}$ where $(x\vee
y)\Rightarrow y\approx\big{(}(x\vee y)\vee y\big{)}^{y}\approx(x\vee y)^{y}$.
∎

##  4 Sectionally and relatively residuated join-semilattices

The aim of this section is to establish a relationship between the structures
defined in the previous section and residuated structures which formalize
certain substructural logics.

We start with definitions of our key concepts relating our implication
algebras with a certain kind of residuation.

######  Definition 4.1.

A sectionally residuated join-semilattice is an ordered quintuple $(S,\vee,$
$(\odot_{x};x\in S),\rightarrow,1)$ such that $(S,\vee,1)$ is a join-
semlattice with top element $1$, for each $x\in S$, $([x,1],\odot_{x},1)$ is a
commutative monoid, $\rightarrow$ is a binary operation on $S$ and for all
$x,y,z,u\in S$ the following hold:

  1. (i)

if $z\leq u\leq x,y$ then $x\odot_{z}y=x\odot_{u}y$ (compatibility condition),

  2. (ii)

if $u\leq x,y,z$ and $x\leq y$ then $x\odot_{u}z\leq y\odot_{u}z$
(monotonicity),

  3. (iii)

$(x\vee z)\odot_{z}(y\vee z)\leq z$ if and only if $x\vee z\leq y\rightarrow
z$ (sectional adjointness),

  4. (iv)

$(x\vee y)\rightarrow y\approx x\rightarrow y$.

The next concept we borrow from our previous paper [11] where, however, it is
introduced for lattices and hence the meet-operation is everywhere defined in
the original definition.

######  Definition 4.2.

A relatively residuated join-semilattice is a partial algebra
$\mathbf{R}=(R,\vee,$ $\odot,$ $\rightarrow,1)$ of type $(2,2,2,0)$ where
$x\odot y$ is defined if and only if $x$ and $y$ have a common lower bound and
in this case $z\leq x\odot y$ for all $z\in R$ with $z\leq x,y$ and
$\rightarrow$ is a binary operation on $R$ such that for all $x,y,z\in R$ the
following hold:

  1. (10)

$(R,\vee,1)$ is a join-semilattice with top element $1$,

  2. (11)

$x\odot 1\approx 1\odot x\approx x$,

  3. (12)

$x\odot y=y\odot x$ whenever $x$ and $y$ have a common lower bound,

  4. (13)

$(x\odot y)\odot z=x\odot(y\odot z)$ whenever $x$, $y$ and $z$ have a common
lower bound,

  5. (14)

$x\leq y$ implies $x\odot z\leq y\odot z$ whenever $x$ and $z$ have a common
lower bound (monotonicity),

  6. (15)

$(x\vee z)\odot(y\vee z)\leq z$ if and only if $x\vee z\leq y\rightarrow z$
(relative adjointness),

  7. (16)

$(x\vee y)\rightarrow y\approx x\rightarrow y$.

$\mathbf{R}$ is called divisible if it satisfies the identity

$(x\vee y)\odot(x\rightarrow y)\approx y.$

In the next theorem we describe several basic properties of relatively
residuated join-semilattices.

######  Theorem 4.3.

Let $\mathbf{S}=(S,\vee,\odot,\rightarrow,1)$ be a relatively residuated join-
semilattice and $a,b,c\in S$. Then the following hold:

  1. (i)

$a\leq b\Leftrightarrow a\rightarrow b=1$,

  2. (ii)

$a\odot b\leq a$,

  3. (iii)

if $a\wedge b$ exists then $a\odot b\leq a\wedge b$,

  4. (iv)

$a\leq b\rightarrow a$,

  5. (v)

$a\odot(a\rightarrow b)\leq(a\vee b)\odot(a\rightarrow b)\leq b$,

  6. (vi)

$a\leq(a\rightarrow b)\rightarrow b$,

  7. (vii)

$a\leq b\Rightarrow b\rightarrow c\leq a\rightarrow c$,

  8. (viii)

$\big{(}(a\rightarrow b)\rightarrow b\big{)}\rightarrow b=a\rightarrow b$.

###### Proof.

  1. (i)

The following are equivalent:

$\displaystyle a$ $\displaystyle\leq b,$ $\displaystyle(1\vee b)\odot(a\vee
b)$ $\displaystyle\leq b,$ $\displaystyle 1\vee b$ $\displaystyle\leq
a\rightarrow b,$ $\displaystyle a\rightarrow b$ $\displaystyle=1.$

  2. (ii)

By commutativity and monotonicity of $\odot$ we have $a\odot b=b\odot a\leq
1\odot a=a$.

  3. (iii)

This follows from (ii) and from commutativity of $\odot$.

  4. (iv)

We have $(a\vee b)\odot a\leq a$ according to (ii). Now the following are
equivalent:

$\displaystyle(a\vee a)\odot(b\vee a)$ $\displaystyle\leq a,$ $\displaystyle
a\vee a$ $\displaystyle\leq b\rightarrow a,$ $\displaystyle a$
$\displaystyle\leq b\rightarrow a.$

  5. (v)

We have $b\leq a\rightarrow b$ according to (iv). Now by (iv) the following
are equivalent:

$\displaystyle(a\rightarrow b)\vee b$ $\displaystyle\leq a\rightarrow b,$
$\displaystyle\big{(}(a\rightarrow b)\vee b\big{)}\odot(a\vee b)$
$\displaystyle\leq b,$ $\displaystyle(a\vee b)\odot(a\rightarrow b)$
$\displaystyle\leq b.$

By monotonicity of $\odot$ we have $a\odot(a\rightarrow b)\leq(a\vee
b)\odot(a\rightarrow b)$.

  6. (vi)

We have $(a\vee b)\odot(a\rightarrow b)\leq b$ according to (v). Now by (iv)
the following are equivalent:

$\displaystyle(a\vee b)\odot\big{(}(a\rightarrow b)\vee b\big{)}$
$\displaystyle\leq b,$ $\displaystyle a\vee b$ $\displaystyle\leq(a\rightarrow
b)\rightarrow b.$

  7. (vii)

Assume $a\leq b$. Then $a\vee c\leq(b\rightarrow c)\rightarrow c$ according to
(iv) and (vi). Now the following are equivalent:

$\displaystyle a\vee c$ $\displaystyle\leq(b\rightarrow c)\rightarrow c,$
$\displaystyle(a\vee c)\odot\big{(}(b\rightarrow c)\vee c\big{)}$
$\displaystyle\leq c,$ $\displaystyle\big{(}(b\rightarrow c)\vee
c\big{)}\odot(a\vee c)$ $\displaystyle\leq c,$ $\displaystyle(b\rightarrow
c)\vee c$ $\displaystyle\leq a\rightarrow c,$ $\displaystyle b\rightarrow c$
$\displaystyle\leq a\rightarrow c.$

  8. (viii)

According to (vi) and (vii) we have

$a\rightarrow b\leq\big{(}(a\rightarrow b)\rightarrow b\big{)}\rightarrow
b\leq a\rightarrow b$

proving (viii).

∎

In next theorem we show how the class of relatively residuated join-
semilattices can be described without using of relative adjointness.

######  Theorem 4.4.

Let $\mathbf{R}=(R,\vee,$ $\odot,$ $\rightarrow,1)$ be a partial algebra of
type $(2,2,2,0)$ where $x\odot y$ is defined if and only if $x$ and $y$ have a
common lower bound and in this case $z\leq x\odot y$ for all $z\in R$ with
$z\leq x,y$ and $\rightarrow$ is a binary operation on $R$ such that for all
$x,y,z\in R$ (10) – (14) and (16) hold. Then $\mathbf{R}$ is a relatively
residuated join-semilattice if and only if it satisfies the following
identities:

  1. (17)

$x\vee z\leq y\rightarrow\Big{(}\big{(}(x\vee z)\odot(y\vee z)\big{)}\vee
z\Big{)}$,

  2. (18)

$x\leq y\rightarrow x$,

  3. (19)

$(x\vee y)\odot(x\rightarrow y)\leq y$.

###### Proof.

Let $a,b,c,d\in R$. If $\mathbf{R}$ is a relatively residuated join-
semilattice then (18) and (19) follow from Theorem 4.3, and if
$d:=\big{(}(a\vee c)\odot(b\vee c)\big{)}\vee c$ then $a\vee d=a\vee c$ and
$b\vee d=b\vee c$ because of (ii) of Theorem 4.3 and (12) whence

$(a\vee d)\odot(b\vee d)=(a\vee c)\odot(b\vee c)\leq d$

by definition of $d$ which implies

$a\vee c=a\vee d\leq b\rightarrow d=b\rightarrow\Big{(}\big{(}(a\vee
c)\odot(b\vee c)\big{)}\vee c\Big{)}$

by (15) showing (17). If, conversely, $\mathbf{R}$ satisfies (17) – (19) then,
if $(a\vee c)\odot(b\vee c)\leq c$ then $a\vee c\leq b\rightarrow c$ according
to (17), and if, conversely, $a\vee c\leq b\rightarrow c$ then

$(a\vee c)\odot(b\vee c)\leq(b\rightarrow c)\odot(b\vee c)=(b\vee
c)\odot(b\rightarrow c)\leq c$

according to (14), (12) and (19) proving (15), thus showing that $\mathbf{R}$
is a relatively residuated join-semilattice. ∎

Theorem 4.4 implies that the class of relatively residuated join-semilattices
is determined by partial identities and hence forms a partial variety.

The following results show that the concepts just defined can be converted
into each other.

######  Theorem 4.5.

  1. (i)

Let $\mathbf{S}=(S,\vee,(\odot_{x};x\in S),\rightarrow,1)$ be a sectionally
residuated join-semilattice and define a partial binary operation $\odot$ on
$S$ as follows: If $x$ and $y$ have a common lower bound $z$ then $x\odot
y:=x\odot_{z}y$. Then $\odot$ is well-defined and
$\mathbb{R}(\mathbf{S}):=(S,\vee,\odot,\rightarrow,1)$ is a relatively
residuated join-semilattice.

  2. (ii)

Let $\mathbf{R}=(R,\vee,\odot,\rightarrow,1)$ be a relatively residuated join-
semilattice and put $\odot_{x}:=\odot|[x,1]$ for all $x\in S$. Then
$\mathbb{S}(\mathbf{R}):=(R,\vee,(\odot_{x};x\in R),\rightarrow,1)$ is a
sectionally residuated join-semilattice.

###### Proof.

(16) coincides with (iv) of Definition 4.1.

  1. (i)

Let $a,b,c,d\in S$. If $c$ and $d$ are common lower bounds of $a$ and $b$ then
also $c\vee d$ is a common lower bound of $a$ and $b$ and hence
$a\odot_{c}b=a\odot_{c\vee d}b=a\odot_{d}b$. This shows that $\odot$ is well-
defined. Moreover, we have $a\odot 1=a\odot_{a}1=a$ and $1\odot
a=1\odot_{a}a=a$. If $a$ and $b$ have a common lower bound $c$ then $a\odot
b=a\odot_{c}b=b\odot_{c}a=b\odot a$. If, finally, $a$, $b$ and $c$ have a
common lower bound $d$ then

$(a\odot b)\odot
c=(a\odot_{d}b)\odot_{d}c=a\odot_{d}(b\odot_{d}c)=a\odot(b\odot c)$

and $a\leq b$ implies $a\odot c=a_{d}c\leq b\odot_{d}c=b\odot c$. Relative
adjointness follows from $(a\vee c)\odot(b\vee c)=(a\vee c)\odot_{c}(b\vee
c)$.

  2. (ii)

Let $a,b,c,d\in R$. Then $([a,1],\odot_{a},1)$ is a commutative monoid.
Moreover, if $c\leq d\leq a,b$ then $a\odot_{c}b=a\odot b=a\odot_{d}b$.
Further, if $d\leq a,b,c$ and $a\leq b$ then $a\odot_{d}c=a\odot c\leq b\odot
c=b\odot_{d}c$. Finally, sectional adjointness follows from $(a\vee
c)\odot_{c}(b\vee c)=(a\vee c)\odot(b\vee c)$.

∎

The next result shows that the conversions described in Theorem 4.5 are
nothing else then a translation from one language into the other.

######  Theorem 4.6.

The correspondence described in Theorem 4.5 is one-to-one.

###### Proof.

If $\mathbf{S}=(S,\vee,(\odot_{x};x\in S),\rightarrow,1)$ is a sectionally
residuated join-semilattice,

$\displaystyle\mathbb{R}(\mathbf{S})$
$\displaystyle=(S,\vee,\odot,\rightarrow,1),$
$\displaystyle\mathbb{S}\big{(}\mathbb{R}(\mathbf{S})\big{)}$
$\displaystyle=(S,\vee,(\circ_{x};x\in S),\rightarrow,1),$

$a,b,c\in S$ and $c\leq a,b$ then $a\odot_{c}b=a\odot b=a\odot_{c}b$ showing
$\mathbb{S}\big{(}\mathbb{R}(\mathbf{S})\big{)}=\mathbf{S}$. If, conversely
$\mathbf{R}=(R,\vee,\odot,\rightarrow,1)$ is a relatively residuated join-
semilattice,

$\displaystyle\mathbb{S}(\mathbf{R})$ $\displaystyle=(R,\vee,(\odot_{x};x\in
R),\rightarrow,1),$
$\displaystyle\mathbb{R}\big{(}\mathbb{S}(\mathbf{R})\big{)}$
$\displaystyle=(R,\vee,\circ,\rightarrow,1),$

$a,b,c\in R$ and $c\leq a,b$ then $a\circ b=a\odot_{c}b=a\odot b$ showing
$\mathbb{R}\big{(}\mathbb{S}(\mathbf{R})\big{)}=\mathbf{R}$. ∎

In the next result we show that non-classical implication semilattices form a
partial subvariety of the class of divisible relatively residuated join-
semilattices.

According to (iii) of Theorem 4.3 $x\odot y\leq x\wedge y$ whenever $x\wedge
y$ exists. The question when $x\odot y=x\wedge y$ is answered in the next
theorem; this equality turns out to hold if $\odot$ is idempotent (and
satisfies one more simple identity ).

######  Theorem 4.7.

The non-classical implication semilattices are exactly the divisible
relatively residuated join-semilattices $(R,\vee,\odot,\rightarrow,1)$
satisfying the following identities:

  1. (i)

$x\odot x\approx x$,

  2. (ii)

$y\leq(x\vee z)\rightarrow\big{(}(x\vee z)\odot(y\vee z)\big{)}$.

###### Proof.

Let $(S,\vee,\wedge,\rightarrow,1)$ be a non-classical implication
semilattice. Then $x\wedge y$ is defined if and only if $x$ and $y$ have a
common lower bound, $z\leq x\wedge y$ whenever $z\leq x,y$, $\rightarrow$ is a
binary operation on $S$ and (10) – (14) of Definition 4.2 hold. Now let
$a,b,c\in S$. First assume $(a\vee c)\wedge(b\vee c)\leq c$ Then $(b\vee
c)\wedge(a\vee c)=c$. Since

$a\leq(b\vee c)\rightarrow\big{(}(b\vee c)\wedge(a\vee c)\big{)}=(b\vee
c)\rightarrow c=b\rightarrow c$

according to (4) and (3) and $c\leq b\rightarrow c$ according to (1), we
conclude $a\vee c\leq b\rightarrow c$. If, conversely, $a\vee c\leq
b\rightarrow c$ then

$(a\vee c)\wedge(b\vee c)=(b\vee c)\wedge(a\vee c)\leq(b\vee
c)\wedge(b\rightarrow c)=c$

by (2). This shows relative adjointness. (16) coincides with (3). Divisibility
of $\mathbf{S}$ follows from (2). Obviously, (i) holds. Moreover, (ii) follows
from (3), and (iii) follows from (4). Conversely, let
$(R,\vee,\odot,\rightarrow,1)$ be a divisible relatively residuated join-
semilattice satisfying identities (i) and (ii). Then $(R,\vee,1)$ is a join-
semilattice with top element $1$ according to (10), $x\odot y$ is defined if
and only if $x$ and $y$ have a common lower bound and $\rightarrow$ is a
binary operation on $R$. Let $a,b,c,d\in R$ with $c\leq a,b$. Because of (14),
(11) and (12) we have $a\odot b\leq 1\odot b=b$ and $a\odot b=b\odot a\leq a$.
If, moreover, $d\leq a,b$ then because of (i), (14) and (12) we have

$d=d\odot d\leq a\odot d=d\odot a\leq b\odot a=a\odot b.$

This shows $a\odot b=a\wedge b$. Finally, (1) follows from (iv) of Theorem
4.3, (2) from divisibility, (3) from (16) and (4) from (ii). ∎

######  Remark 4.8.

In Theorem 4.7 identity (i) can be replaced by condition (i’)

  1. (i’)

$x\wedge y$ exists if and only if $x$ and $y$ have a common lower bound.

This can be seen as follows: Let $(R,\vee,\odot,\rightarrow,1)$ be a
relatively residuated join-semilattice satisfying (i’) and $a,b\in R$ having a
common lower bound. Then $a\wedge b$ exists by (i’) and $a\wedge b$ is a
common lower bound of $a$ and $b$ whence $a\wedge b\leq a\odot b$. But $a\odot
b\leq a\wedge b$ by (iii) of Theorem 4.3 which shows $a\odot b=a\wedge b$.

######  Example 4.9.

The join-semilattice depicted in Figure 1

$a$$b$$1$$c$$d$Fig. 1

is a join-semilattice whose sections are pseudocomplemented lattices and the
operation tables for $\rightarrow$ and $\wedge$ look as follows:

$\begin{array}[]{c|ccccc}\rightarrow&a&b&c&d&1\\\ \hline\cr a&1&1&c&d&1\\\
b&a&1&c&d&1\\\ c&a&b&1&1&1\\\ d&a&b&c&1&1\\\
1&a&b&c&d&1\end{array}\quad\quad\quad\begin{array}[]{c|ccccc}\wedge&a&b&c&d&1\\\
\hline\cr a&a&a&-&-&a\\\ b&a&b&-&-&b\\\ c&-&-&c&c&c\\\ d&-&-&c&d&d\\\
1&a&b&c&d&1\end{array}$

It is easy to check adjointness in the sense that whenever $x\wedge y$ is
defined we have $x\wedge y\leq z$ if and only $x\leq y\rightarrow z$. It
should be mentioned that $\rightarrow$ is a proper extension of the
pseudocomplementation within sections. Namely, the pseudocomplement of $b$
within $[a,1]$ is $b\rightarrow a=a$. On the other hand, $b\notin[d,1]$, but
$b\rightarrow d=d$.

##  5 The variety of non-classical implication algebras

Up to now, we investigated algebras where the operations $\vee$ and
$\rightarrow$ were everywhere defined, but $\wedge$ or $\odot$ were only
partial operations defined just in the case when both the operands belong to
the same section. To avoid this difficulty, we introduce a new ternary
operation $r(x,y,z)$ which is everywhere defined and replaces the partial
operation $\wedge$.

######  Definition 5.1.

An I-algebra is an algebra $(A,\vee,r,\rightarrow,1)$ of type $(2,3,2,0)$ such
that $(A,\vee,1)$ is a join-semilattice with top element $1$ and the following
identities are satisfied:

  1. (1’)

$y\leq x\rightarrow y$,

  2. (2’)

$r(x,x\rightarrow y,y)\approx y$,

  3. (3’)

$(x\vee y)\rightarrow y\approx x\rightarrow y$,

  4. (4’)

$y\leq(x\vee z)\rightarrow r(x,y,z)$,

  5. (5’)

$r(x,y,z)\leq x\vee z$,

  6. (6’)

$r(x,y,z)\leq y\vee z$,

  7. (7’)

$r(x,x\vee y,z)\approx x\vee z$,

  8. (8’)

$r(x,y,z)\approx r(x\vee z,y\vee z,z)$,

  9. (9’)

$z\leq r(x,y,z)$,

  10. (10’)

$r(u,r(x,y,z),z)\approx r(r(u,x,z),r(u,y,z),z)$.

Considering a non-classical implication semilattice as introduced in
Definition 3.2, we can define a ternary operation $r(x,y,z)$ as follows:
$r(x,y,z):=(x\vee z)\wedge(y\vee z)$. For every non-classical implication
semilattice $\mathbf{I}=(I,\vee,\wedge,\rightarrow,1)$ put
$\mathbb{A}(\mathbf{I}):=(I,\vee,r,\rightarrow,1)$.

The next theorem shows a conversion of non-classical implication join-
semilattices into I-algebras.

######  Theorem 5.2.

Let $\mathbf{I}=(I,\vee,\wedge,\rightarrow,1)$ be a non-classical implication
semilattice. Then $\mathbb{A}(\mathbf{I})$ is an I-algebra.

###### Proof.

Let $\mathbb{A}(\mathbf{I})=(I,\vee,r,\rightarrow,1)$ and $a,b,c,d\in I$.
Then, clearly, $r(x,x,y)\approx x\vee y\approx r(y,y,x)$.

  1. (1’)

coincides with (1).

  2. (2’)

$r(a,a\rightarrow b,b)=(a\vee b)\wedge\big{(}(a\rightarrow b)\vee
b\big{)}=(a\vee b)\wedge(a\rightarrow b)=b$ according to (1) and (2).

  3. (3’)

coincides with (3).

  4. (4’)

$b\leq(a\vee c)\rightarrow\big{(}(a\vee c)\wedge(b\vee c)\big{)}=(a\vee
c)\rightarrow r(a,b,c)$ according to (4).

  5. (5’)

$r(a,b,c)=(a\vee c)\wedge(b\vee c)\leq a\vee c$

  6. (6’)

$r(a,b,c)=(a\vee c)\wedge(b\vee c)\leq b\vee c$

  7. (7’)

$r(a,a\vee b,c)=(a\vee c)\wedge\big{(}(a\vee b)\vee c\big{)}=a\vee c$

  8. (8’)

$r(a,b,c)=(a\vee c)\wedge(b\vee c)=\big{(}(a\vee c)\vee
c\big{)}\wedge\big{(}(b\vee c)\vee c\big{)}=r(a\vee c,b\vee c,c)$

  9. (9’)

$c\leq(a\vee c)\wedge(b\vee c)=r(a,b,c)$

  10. (10’)

$\displaystyle r(d,r(a,b,c),c)$ $\displaystyle=(d\vee
c)\wedge\Big{(}\big{(}(a\vee c)\wedge(b\vee c)\big{)}\vee c\Big{)}=(d\vee
c)\wedge\big{(}(a\vee c)\wedge(b\vee c)\big{)}=$ $\displaystyle=(d\vee
c)\wedge(a\vee c)\wedge(b\vee c)=$ $\displaystyle=\big{(}(d\vee c)\wedge(a\vee
c)\big{)}\wedge\big{(}(d\vee c)\wedge(b\vee c)\big{)}=$
$\displaystyle=\Big{(}\big{(}(d\vee c)\wedge(a\vee c)\big{)}\vee
c\Big{)}\wedge\Big{(}\big{(}(d\vee c)\wedge(b\vee c)\big{)}\vee c\Big{)}=$
$\displaystyle=r\big{(}r(d,a,c),r(d,b,c),c\big{)}$

∎

For every I-algebra $\mathbf{A}=(A,\vee,r,\rightarrow,1)$ put
$\mathbb{J}(\mathbf{A}):=(A,\vee,\wedge,\rightarrow,1)$ where $x\wedge
y:=r(x,y,z)$ for all $x,y,z\in A$ with $z\leq x,y$. That
$\mathbb{J}(\mathbf{A})$ is well-defined follows from the next theorem.

######  Theorem 5.3.

Let $\mathbf{A}=(A,\vee,r,\rightarrow,1)$ be an I-algebra. Then
$\mathbb{J}(\mathbf{A})$ is a well-defined non-classical implication
semilattice.

###### Proof.

Let $\mathbb{J}(\mathbf{A})=(A,\vee,\wedge,\rightarrow,1)$ and $a,b,c,d\in A$.
First we prove $r(a,b,c)=(a\vee c)\wedge(b\vee c)$. Because of (5’) and (6’)
we have $r(a,b,c)\leq a\vee c,b\vee c$. Now assume $d\leq a\vee c,b\vee c$.
Then

$\displaystyle d$ $\displaystyle\leq d\vee c=(d\vee c)\vee c=r\big{(}d\vee
c,(d\vee c)\vee c,c\big{)}=r\big{(}(d\vee c)\vee c,(d\vee c)\vee c,c\big{)}=$
$\displaystyle=r\Big{(}r\big{(}d\vee c,(d\vee c)\vee(a\vee
c),c\big{)},r\big{(}d\vee c,(d\vee c)\vee(b\vee c),c\big{)},c\Big{)}=$
$\displaystyle=r\big{(}r(d\vee c,a\vee c,c),r(d\vee c,b\vee
c,c),c\big{)}=r\big{(}r(d,a,c),r(d,b,c),c\big{)}=r\big{(}d,r(a,b,c),c\big{)}\leq$
$\displaystyle\leq r(a,b,c)\vee c=r(a,b,c)$

because of (7’), (8’), (10’), (6’) and (9’). This shows $r(a,b,c)=(a\vee
c)\wedge(b\vee c)$. Hence $a\wedge b=\inf(a,b)$ whenever there exists a common
lower bound of $a$ and $b$ and therefore $\wedge$ and $\mathbb{J}(\mathbf{A})$
are well-defined.

  1. (1)

coincides with (1’).

  2. (2)

$(a\vee b)\wedge(a\rightarrow b)=r(a,a\rightarrow b,b)=b$ because of (1’) and
(2’).

  3. (3)

coincides with (3’).

  4. (4)

$b\leq(a\vee c)\rightarrow r(a,b,c)=(a\vee c)\rightarrow\big{(}(a\vee
c)\wedge(b\vee c)\big{)}$ by (4’).

∎

######  Theorem 5.4.

The correspondence described in Theorems 5.2 and 5.3 is one-to-one.

###### Proof.

If $\mathbf{I}=(I,\vee,\wedge,\rightarrow,1)$ is a non-classical implication
semilattice and

$\displaystyle\mathbb{A}(\mathbf{I})$
$\displaystyle=(I,\vee,r,\rightarrow,1),$
$\displaystyle\mathbb{J}\big{(}\mathbb{A}(\mathbf{I})\big{)}$
$\displaystyle=(I,\vee,\cap,\rightarrow,1)$

then $x\cap y=r(x,y,z)=(x\vee z)\wedge(y\vee z)=x\wedge y$ for all $x,y,z\in
I$ with $z\leq x,y$ and hence
$\mathbb{J}\big{(}\mathbb{A}(\mathbf{I})\big{)}=\mathbf{I}$. If, conversely,
$\mathbf{A}=(A,\vee,r,\rightarrow,1)$ is an I-algebra and

$\displaystyle\mathbb{J}(\mathbf{A})$
$\displaystyle=(A,\vee,\wedge,\rightarrow,1),$
$\displaystyle\mathbb{A}\big{(}\mathbb{J}(\mathbf{A})\big{)}$
$\displaystyle=(A,\vee,R,\rightarrow,1)$

then $R(x,y,z)\approx(x\vee z)\wedge(y\vee z)\approx r(x\vee z,y\vee
z,z)\approx r(x,y,z)$ according to (8’) and hence
$\mathbb{A}\big{(}\mathbb{J}(\mathbf{A})\big{)}=\mathbf{A}$. ∎

The advantage of using a ternary operation $r(x,y,z)$ instead of the partial
binary operation $\wedge$ is that now we have an algebra with everywhere
defined operations which is determined by a finite set of identities. Hence
these algebras form a variety $\mathcal{V}$.

Recall that an algebra $\mathbf{A}$ is called

  * •

$3$-permutable if $\Theta\circ\Phi\circ\Theta=\Phi\circ\Theta\circ\Phi$ for
all $\Theta,\Phi\in\operatorname{Con}\mathbf{A}$,

  * •

congruence distributive if the congruence lattice
$\operatorname{\mathbf{Con}}\mathbf{A}$ of $\mathbf{A}$ is distributive.

An algebra $\mathbf{A}$ with an equationally definable constant $1$ is called
weakly regular if $\Theta,\Phi\in\operatorname{Con}\mathbf{A}$ and
$[1]\Theta=[1]\Phi$ imply $\Theta=\Phi$. A variety is called $3$-permutable or
congruence distributive if every of its members has the respective property. A
variety with an equationally definable constant $1$ is called weakly regular
if every of its members has this property. A variety is $3$-permutable (cf.
[8], Theorem 3.1.18) if and only if there exist ternary terms $t_{1},t_{2}$
satisfying

$\displaystyle t_{1}(x,y,y)$ $\displaystyle\approx x,$ $\displaystyle
t_{1}(x,x,y)$ $\displaystyle\approx t_{2}(x,y,y),$ $\displaystyle
t_{2}(x,x,y)$ $\displaystyle\approx y.$

A variety is congruence distributive (cf. [8], Theorem 3.2.2) if and only if
there exists a positive integer $n$ and there exist ternary terms
$t_{0},\ldots,t_{n}$ (so-called Jónsson terms) satisfying

$\displaystyle t_{0}(x,y,z)$ $\displaystyle\approx x,$ $\displaystyle
t_{i}(x,x,y)$ $\displaystyle\approx t_{i+1}(x,x,y)\text{ for all even
}i\in\\{0,\ldots,n-1\\},$ $\displaystyle t_{i}(x,y,y)$ $\displaystyle\approx
t_{i+1}(x,y,y)\text{ for all odd }i\in\\{1,\ldots,n-1\\},$ $\displaystyle
t_{n}(x,y,z)$ $\displaystyle\approx z,$ $\displaystyle t_{i}(x,y,x)$
$\displaystyle\approx x\text{ for all }i\in\\{1,\ldots,n-1\\}.$

A variety with an equationally definable constant $1$ is weakly regular (cf.
[8], Theorem 6.4.3) if and only if there exists a positive integer $n$ and
there exist binary terms $t_{1},\ldots,t_{n}$ such that
$t_{1}(x,y)=\cdots=t_{n}(x,y)=1$ is equivalent to $x=y$.

######  Theorem 5.5.

The variety $\mathcal{V}$ of I-algebras is $3$-permutable, congruence
distributive and weakly regular.

###### Proof.

Let $\mathbf{A}=(A,\vee,r,\rightarrow,1)$ be an I-algebra and
$\mathbb{J}(\mathbf{A})=(A,\vee,\wedge,\rightarrow,1)$.
If we put

$\displaystyle t_{1}(x,y,z)$ $\displaystyle:=r(z,y\rightarrow x,x),$
$\displaystyle t_{2}(x,y,z)$ $\displaystyle:=r(x,y\rightarrow z,z)$

then

$\displaystyle t_{1}(x,y,z)$ $\displaystyle\approx(z\vee
x)\wedge\big{(}(y\rightarrow x)\vee x\big{)}\approx(z\vee
x)\wedge\big{(}y\rightarrow x)\text{ by (1)},$ $\displaystyle t_{2}(x,y,z)$
$\displaystyle\approx(x\vee z)\wedge\big{(}(y\rightarrow z)\vee
z\big{)}\approx(x\vee z)\wedge\big{(}y\rightarrow z)\text{ by (1)},$
$\displaystyle t_{1}(x,y,y)$ $\displaystyle\approx(y\vee x)\wedge(y\rightarrow
x)\approx x\text{ by (2)},$ $\displaystyle t_{1}(x,x,y)$
$\displaystyle\approx(y\vee x)\wedge\big{(}x\rightarrow x)\approx(x\vee
y)\wedge 1\approx(x\vee y)\wedge\big{(}y\rightarrow y)\approx
t_{2}(x,y,y)\text{ by (5)},$ $\displaystyle t_{2}(x,x,y)$
$\displaystyle\approx(x\vee y)\wedge\big{(}x\rightarrow y)\approx y\text{ by
(2)}.$

and hence $\mathcal{V}$ is $3$-permutable.
If we put

$\displaystyle t_{0}(x,y,z)$ $\displaystyle:=x,$ $\displaystyle t_{1}(x,y,z)$
$\displaystyle:=r(z,y,x),$ $\displaystyle t_{2}(x,y,z)$
$\displaystyle:=r(x,y\rightarrow z,z),$ $\displaystyle t_{3}(x,y,z)$
$\displaystyle:=z$

then

$\displaystyle t_{1}(x,y,z)$ $\displaystyle\approx(z\vee x)\wedge(y\vee x),$
$\displaystyle t_{2}(x,y,z)$ $\displaystyle\approx(x\vee
z)\wedge\big{(}(y\rightarrow z)\vee z\big{)}\approx(x\vee
z)\wedge(y\rightarrow z)\text{ by (1)},$ $\displaystyle t_{0}(x,x,y)$
$\displaystyle\approx x\approx(x\vee y)\wedge x\approx(y\vee x)\wedge(x\vee
x)\approx t_{1}(x,x,y),$ $\displaystyle t_{1}(x,y,y)$
$\displaystyle\approx(y\vee x)\wedge(y\vee x)\approx x\vee y\approx(x\vee
y)\wedge 1\approx(x\vee y)\wedge(y\rightarrow y)\approx t_{2}(x,y,y)$
$\displaystyle\hskip 17.07164pt\text{by (5)},$ $\displaystyle t_{2}(x,x,y)$
$\displaystyle\approx(x\vee y)\wedge(x\rightarrow y)\approx y\approx
t_{3}(x,x,y)\text{ by (2)},$ $\displaystyle t_{1}(x,y,x)$
$\displaystyle\approx(x\vee x)\wedge(y\vee x)\approx x\wedge(x\vee y)\approx
x,$ $\displaystyle t_{2}(x,y,x)$ $\displaystyle\approx(x\vee
x)\wedge(y\rightarrow x)\approx x\wedge(y\rightarrow x)\approx x\text{ by
(1)}$

and hence $\mathcal{V}$ is congruence distributive.
Finally, if we put

$\displaystyle t_{1}(x,y)$ $\displaystyle:=x\rightarrow y,$ $\displaystyle
t_{2}(x,y)$ $\displaystyle:=y\rightarrow x$

then $t_{1}(x,y)=t_{2}(x,y)=1$ is equivalent to $x\leq y$ and $y\leq x$
according to Theorem 5.3 and (5), i.e. to $x=y$ and hence $\mathcal{V}$ is
weakly regular. ∎

##  6 Algebras assigned to relatively residuated join-
-semilattices

Similarly as for non-classical implication semilattices, we are going to show
that also relatively residuated join-semilattices can be converted into
algebras with everywhere defined operations by using of a certain ternary term
$q(x,y,z)$ which replaces the partial operation $\odot$.

######  Definition 6.1.

An R-algebra is an algebra $(R,\vee,q,\rightarrow,1)$ of type $(2,3,2,0)$ such
that $(R,\vee,1)$ is a join-semilattice with top element $1$ and the following
conditions are satisfies for all $x,y,z,u\in R$:

  1. (20)

$z\leq q(x,y,z)$,

  2. (21)

$q(z\vee u\vee x,z\vee u\vee y,z)\approx q(z\vee u\vee x,z\vee u\vee y,z\vee
u)$,

  3. (22)

$q(x,1,x)\approx q(1,x,x)\approx x$,

  4. (23)

$q(x,y,z)\approx q(y,x,z)$,

  5. (24)

$q(q(x,y,u),z,u)\approx q(x,q(y,z,u),u)$,

  6. (25)

$q(x,z,u)\leq q(x\vee y,z,u)$,

  7. (26)

$x\vee z\leq y\rightarrow\big{(}q(x,y,z)\vee z\big{)}$,

  8. (27)

$x\leq y\rightarrow x$,

  9. (28)

$q(x,x\rightarrow y,y)\leq y$,

  10. (29)

$q(x,y,z)\approx q(x\vee z,y\vee z,z)$,

  11. (30)

$(x\vee y)\rightarrow y\approx x\rightarrow y$.

The following result is an immediate consequence of Definition 6.1.

######  Corollary 6.2.

The class $\mathcal{W}$ of R-algebras forms a variety.

Considering a relatively residuated join-semilattice as introduced in
Definition 4.2, we can define a ternary operation $q(x,y,z)$ as follows:
$q(x,y,z):=(x\vee z)\odot(y\vee z)$. For every relatively residuated join-
semilattice $\mathbf{R}=(R,\vee,\odot,\rightarrow,1)$ put
$\mathbb{B}(\mathbf{R}):=(R,\vee,q,\rightarrow,1)$.

######  Theorem 6.3.

Let $\mathbf{R}=(R,\vee,\odot,\rightarrow,1)$ be a relatively residuated join-
semilattice. Then $\mathbb{B}(\mathbf{R})$ is an R-algebra.

###### Proof.

Let $\mathbb{B}(\mathbf{R})=(R,\vee,q,\rightarrow,1)$ and $a,b,c,d\in R$. We
have

  1. (20)

$c\leq(a\vee c)\odot(b\vee c)=q(a,b,c)$.

  2. (21)

$q(c\vee d\vee a,c\vee d\vee b,c)=(c\vee d\vee a)\odot(c\vee d\vee b)=q(c\vee
d\vee a,c\vee d\vee b,c\vee d)$

  3. (22)

$q(a,1,a)=(a\vee a)\odot(1\vee a)=a\odot 1=a$ and $q(1,a,a)=(1\vee
a)\odot(a\vee a)=1\odot a=a$.

  4. (23)

$(a,b,c)=(a\vee c)\odot(b\vee c)=(b\vee c)\odot(a\vee c)=q(b,a,c)$.

  5. (24)

$q(q(a,b,d)c,d)=\Big{(}\big{(}(a\vee d)\odot(b\vee d)\big{)}\vee
d\Big{)}\odot(c\vee d)=\big{(}(a\vee d)\odot(b\vee d)\big{)}\odot(c\vee
d)=(a\vee d)\odot\big{(}(b\vee d)\odot(c\vee d)\big{)}=(a\vee
d)\odot\Big{(}\big{(}(b\vee d)\odot(c\vee d)\big{)}\vee d\Big{)}$.

  6. (25)

$q(a,c,d)=(a\vee d)\odot(c\vee d)\leq\big{(}(a\vee b)\vee d)\odot(c\vee
d)=q(a\vee b,c,d)$.

  7. (26)

According to (17) we have

$a\vee c\leq b\rightarrow\Big{(}\big{(}(a\vee c)\odot(b\vee c)\big{)}\vee
c\Big{)}=b\rightarrow\big{(}q(a,b,c)\vee c\big{)}.$

  8. (27)

coincides with (18).

  9. (28)

According to (18) and (19) we have

$q(a,a\rightarrow b,b)=(a\vee b)\odot\big{(}(a\rightarrow b)\vee
b\big{)}=(a\vee b)\odot(a\rightarrow b)\leq b.$

  10. (29)

$q(a,b,c)=(a\vee c)\odot(b\vee c)=\big{(}(a\vee c)\vee
c\big{)}\odot\big{(}(b\vee c)\vee c\big{)}=q(a\vee c,b\vee c,c)$.

  11. (30)

coincides with (16).

∎

For every R-algebra $\mathbf{R}=(R,\vee,q,\rightarrow,1)$ put
$\mathbb{Q}(\mathbf{R}):=(R,\vee,\odot,\rightarrow,1)$ where $x\odot
y:=q(x,y,z)$ for all $x,y,z\in R$ with $z\leq x,y$. That
$\mathbb{Q}(\mathbf{R})$ is well-defined follows from the next theorem.

######  Theorem 6.4.

Let $\mathbf{R}=(R,\vee,q,\rightarrow,1)$ be an R-algebra. Then
$\mathbb{Q}(\mathbf{R})$ is a well-defined relatively residuated join-
semilattice.

###### Proof.

Let $\mathbb{Q}(\mathbf{R})=(R,\vee,\odot,\rightarrow,1)$ and $a,b,c,d\in R$.
Then $\odot$ is well-defined since in case $c,d\leq a,b$ we have

$q(a,b,c)=q(a\vee c\vee d,b\vee c\vee d,c)=q(a\vee c\vee d,b\vee c\vee d,c\vee
d)=q(a\vee c\vee d,b\vee c\vee d,d)=q(a,b,d)$

by (21). Then because of (20), $c\leq q(a,b,c)=a\odot b$ whenever $c\leq a,b$.

  1. (10)

follows from the definition of an R-algebra.

  2. (11)

$a\odot 1=q(a,1,a)=a$ and $1\odot a=q(1,a,a)=a$ by (22).

  3. (12)

Because of (23) we have $a\odot b=q(a,b,c)=q(b,a,c)=b\odot a$ whenever $c\leq
a,b$.

  4. (13)

Because of (24) we have $(a\odot b)\odot
c=q(q(a,b,d),c,d)=q(a,q(b,c,d),d)=a\odot(b\odot c)$ whenever $d\leq a,b,c$.

  5. (14)

Because of (25) we have $a\odot c=q(a,c,d)\leq q(a\vee b,c,d)=q(b,c,d)=b\odot
c$ whenever $d\leq a\leq b$ and $d\leq c$.

  6. (16)

coincides with (30).

  7. (17)

Because of (26) and (29) we have

$a\vee c\leq b\rightarrow\big{(}q(a,b,c)\vee
c\big{)}=b\rightarrow\big{(}q(a\vee c,b\vee c,c)\vee
c\big{)}=b\rightarrow\big{(}(a\vee c)\odot(b\vee c)\big{)}\vee c.$

  8. (18)

coincides with (27).

  9. (19)

Because of (27), (29) and (28) we have

$(a\vee b)\odot(a\rightarrow b)=(a\vee b)\odot\big{(}(a\rightarrow b)\vee
b\big{)}=q\big{(}a\vee b,(a\rightarrow b)\vee b,b)=q(a,a\rightarrow b,b)\leq
b.$

∎

######  Theorem 6.5.

The correspondence described in Theorems 6.3 and 6.4 is one-to-one.

###### Proof.

If $\mathbf{R}=(R,\vee,\odot,\rightarrow,1)$ is a relatively residuated join-
semilattice and

$\displaystyle\mathbb{B}(\mathbf{R})$
$\displaystyle=(R,\vee,q,\rightarrow,1),$
$\displaystyle\mathbb{Q}\big{(}\mathbb{B}(\mathbf{R})\big{)}$
$\displaystyle=(R,\vee,\otimes,\rightarrow,1)$

then $x\otimes y=q(x,y,z)=(x\vee z)\odot(y\vee z)=x\odot y$ for all $x,y,z\in
I$ with $z\leq x,y$ and hence
$\mathbb{J}\big{(}\mathbb{A}(\mathbf{I})\big{)}=\mathbf{I}$. If, conversely,
$\mathbf{R}=(R,\vee,q,\rightarrow,1)$ is an R-algebra and

$\displaystyle\mathbb{Q}(\mathbf{R})$
$\displaystyle=(R,\vee,\odot,\rightarrow,1),$
$\displaystyle\mathbb{B}\big{(}\mathbb{Q}(\mathbf{R})\big{)}$
$\displaystyle=(R,\vee,Q,\rightarrow,1)$

then $Q(x,y,z)\approx(x\vee z)\odot(y\vee z)\approx q(x\vee z,y\vee
z,z)\approx q(x,y,z)$ according to (29) and hence
$\mathbb{B}\big{(}\mathbb{Q}(\mathbf{R})\big{)}=\mathbf{R}$. ∎

By Theorems 6.3, 6.4 and 6.5 the variety $\mathcal{W}$ of R-algebras is
equivalent to the class of relatively residuated join-semilattices.

######  Theorem 6.6.

The variety $\mathcal{W}$ of R-algebras is weakly regular and its subvariety
determined by the additional identity $q(x,x\rightarrow y,y)\approx y$ is
$3$-permutable.

###### Proof.

The proof is analogous to that of Theorem 5.5 and uses also (i) of Theorem
4.3. Since $y\leq x\rightarrow y$ according to (18), the identity
$q(x,x\rightarrow y,y)\approx y$ is equivalent to the identity

$(x\vee y)\odot(x\rightarrow y)\approx y$

which is nothing else than divisibility of the corresponding relatively
residuated join-semilattice. ∎

Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

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Authors’ addresses:

Ivan Chajda
Palacký University Olomouc
Faculty of Science
Department of Algebra and Geometry
17\. listopadu 12
771 46 Olomouc
Czech Republic
<EMAIL_ADDRESS>
Helmut Länger
TU Wien
Faculty of Mathematics and Geoinformation
Institute of Discrete Mathematics and Geometry
Wiedner Hauptstraße 8-10
1040 Vienna
Austria, and
Palacký University Olomouc
Faculty of Science
Department of Algebra and Geometry
17\. listopadu 12
771 46 Olomouc
Czech Republic
<EMAIL_ADDRESS>