# Two Galois connections for left quasigroups

Marco Bonatto  Dipartimento di matematica e informatica - UNIFE
<EMAIL_ADDRESS>
###### Abstract.

We investigate two Galois connection between the congruence lattice and the
lattice of subgroups of the displacement group of left quasigroups. Such
connections were already studied for racks and quandles. We introduce the
class of left quasigroups having congruence determined by subgroups (resp.
orbits) and we extend a known result for quandles.

## Introduction

Left quasigroups are rather combinatorial objects, nevertheless several
algebraic structures arising in different ares of mathematics have an
underlying left quasigroup structure. For instance racks and quandles coming
from low dimensional topology [Joy82, Mat82] and the study of Hopf algebras
[AG03] are examples of highly structured left quasigroups.

A tool for the study of racks is the left multiplication group and its
subgroups. In particular the displacement group reflects a lot of properties
of racks. In particular properties as abelianness and nilpotency in the sense
of commutator theory [H.76, FM87] are completely determined by the
correspondent properties of the displacement group [BS21].

In [BS21], a monotone Galois connection between the lattice of congruences and
the lattice of normal subgroups of the left multiplication group of racks has
been defined by the pair of operators $(\mathrm{Dis}_{*},\mathrm{con}_{*})$.
If this Galois connection provides an isomorphism between the two lattices we
said that a rack has the congruence determined by subgroups (shortly, CDSg).
We also discovered a second Galois connection defined by the pair of mappings
$(\mathrm{Dis}^{*},\mathcal{O}_{*})$ and later on in [Bon21] we introduced the
same pair of operators between the lattice of congruences of left quasigroups
and the lattice of admissible subgroups (in the case of racks, admissible
subgroups and normal subgroup of the left multiplication group are the same
thing). We say that a left quasigroup has the congruence determined by orbits
(shortly, CDOs) whenever this second Galois connection provides an isomorphism
between the two lattices.

In this paper we investigate the relation between these two Galois connections
in the setting of left quasigroups. In particular, left quasigroups having
CDSg and CDOs are compared in Theorem 2.7. Some results in this direction were
already obtained for racks in [Bon20]. One of the main results of the paper is
Theorem 3.8 in which we extend [Bon20, Proposition 3.17] to idempotent left
quasigroups of arbitrary cardinality.

In Section 1 we collect all the basics about left quasigroups and the
interplay between congruences and admissible subgroups. Moreover we introduce
the pairs of operators $(\mathrm{Dis}^{*},\mathcal{O}_{*})$ and
$(\mathrm{Dis}_{*},\mathrm{con}_{*})$ and how they relate with the Cayley
kernel and the property of being faithful. Section 2 is dedicated to the
Galois connections defined by the operators introduced in the previous section
and to left quasigroups having CDSg and CDOs. The last section is about
nilpotent left quasigroups having CDSg (we also provide some basic notions of
commutator theory and central extensions).

##  1\. Preliminary results

###  1.1. Left quasigroups

A left quasigroup is a binary algebraic structure $(Q,\cdot,\backslash)$ such
that the following identities hold:

$x\cdot(x\backslash y)\approx y\approx x\backslash(x\cdot y).$

We will denote the $\cdot$ operation just by juxtaposition in the rest of the
paper. The left and right multiplication mappings of $Q$ are defined as

$L_{x}:y\mapsto xy,\quad R_{x}:y\mapsto yx$

for every $x\in Q$. According to the axioms above, the map $L_{x}$ is a
permutation for every $x\in Q$ and so we can define the left multiplication
group of $Q$ as $\mathrm{LMlt}(Q)=\langle L_{x},\,x\in Q\rangle$. Note that
the $\backslash$ operation is defined as $x\backslash y=L_{x}^{-1}(y)$ for
every $x,y\in Q$, so we ofter specify just the $\cdot$ operation for left
quasigroups, from which it is usually easy to get the left multiplication
mappings and their inverses.

Right quasigroups can be defined analogously as binary algebraic structure
$(Q,\cdot,/)$ satisfying

$(y\cdot x)/x\approx y\approx(y/x)\cdot x.$

The right multiplication mappings of right quasigroups are bijections.

The set of idempotent elements of $Q$ is $E(Q)=\\{x\in Q\,:\,xx=x\\}$. We say
that $Q$ is:

  * (i)

idempotent if $Q=E(Q)$, i.e. the identity $xx\approx x$ holds in $Q$;

  * (ii)

a rack if the identity $x(yz)\approx(xy)(xz)$ holds (or equivalently
$L_{x}\in\mathrm{Aut}(Q,\cdot)$ for every $x\in Q$). Idempotent racks are
called quandles;

  * (iii)

latin if the right multiplications are bijective.

Latin left quasigroups are essentially reducts of quasigroups, namely binary
algebraic structures with three binary operations $(Q,\cdot,\backslash,/)$,
such that $(Q,\cdot,\backslash)$ is a left quasigroup and $(Q,\cdot,/)$ is a
right quasigroup. The two type of structures have different signatures
effecting congruences and subalgebras if the underlying set has infinite
cardinality. In this paper we consider latin left quasigroups with signature
$\\{\cdot,\backslash\\}$.

A term in the language of left quasigroups $\\{\cdot,\backslash\\}$ is either
a variable or an expression $t_{1}\cdot t_{2}$, $t_{1}\backslash t_{2}$, where
$t_{1},t_{2}$ are terms. A Malt’sev term is a ternary term $m$ such that
$m(x,y,y)\approx x\approx m(y,y,x)$ holds. We say that a left quasigroup $Q$
is Malt’sev if the variety generated by $Q$ has a Malt’sev term. Given a left
quasigroup $(Q,\cdot,\backslash)$ any $n$-ary term $t$ provides a map
$t^{Q}:Q^{n}\longrightarrow Q$ called term operation.

For further details on the universal algebraic definitions of terms and
varieties of algebras see [Ber12].

###  1.2. Congruences and admissible subgroups

In this section we collect all the results on congruences and admissible
subgroups we are using in the following. For further details see [Bon21,
Bon22a].

Let $Q$ be a set, we denote the lattice of equivalence relations on $Q$ as
$Equiv(Q)$. Given $\alpha\in Equiv(Q)$ we denote the quotient set with respect
to $\alpha$ by $Q/\alpha$ and the class of $x$ (with respect to $\alpha$) by
$[x]_{\alpha}$ (we often omit the subscrit and we write just $[x]$).

Let $(Q,\cdot,\backslash)$ be a left quasigroup. We can define the following
operators associating to every equivalence a pair of subgroups of
$\mathrm{LMlt}(Q)$:

$\displaystyle\mathrm{Dis}_{*}:\alpha\mapsto\mathrm{Dis}_{\alpha}$
$\displaystyle=\langle
hL_{x}L_{y}^{-1}h^{-1},\,x\,\alpha\,y,\,h\in\mathrm{LMlt}(Q)\rangle,$
$\displaystyle\mathrm{Dis}^{*}:\alpha\mapsto\mathrm{Dis}^{\alpha}$
$\displaystyle=\\{h\in\mathrm{LMlt}(Q)\,:\,h(x)\,\alpha\,x\,\text{ for all
}x\in Q\\}.$

The two operators $\mathrm{Dis}_{*},\mathrm{Dis}^{*}$ are clearly monotone.

The first of the two groups is called the displacement group relative to
$\alpha$ and it is normal in $\mathrm{LMlt}(Q)$ by definition. In particular,
we denote $\mathrm{Dis}_{1_{Q}}$ just by $\mathrm{Dis}(Q)$ and we call it the
displacement group of $Q$. According to [Bon21, Lemma 1.4], we have

$\displaystyle\mathrm{LMlt}(Q)$ $\displaystyle=\mathrm{Dis}(Q)\langle
L_{x}\rangle\,\text{ for every }\,x\in Q,$ $\displaystyle\mathrm{Dis}(Q)$
$\displaystyle=\\{L_{x_{1}}^{k_{1}}\ldots L_{x_{n}}^{k_{n}}\,:\,x_{1},\ldots
x_{n}\in Q,\,n\in\mathbb{N},\,k_{i}\in\mathbb{Z},\,\sum_{j}k_{j}=0\\}.$

A congruence on $Q$ is an equivalence relation $\alpha$ such that
$xy\,\alpha\,zt$ and $x\backslash y\,\alpha\,z\backslash t$ provided
$x\,\alpha\,z$ and $y\,\alpha\,t$. We denote the lattice of congruences of $Q$
as $Con(Q)$. Given $\alpha\in Con(Q)$, the quotient set $Q/\alpha$ is endowed
with a well-defined left quasigroup structure defined by

$[x]_{\alpha}\cdot[y]_{\alpha}=[x\cdot
y]_{\alpha},\,[x]_{\alpha}\backslash[y]_{\alpha}=[x\backslash y]_{\alpha}$

for every $[x]_{\alpha},[y]_{\alpha}\in Q/\alpha$. The canonical map
$x\mapsto[x]_{\alpha}$ provides a surjective morphism of left quasigroups from
$Q$ to $Q/\alpha$. On the other hand, given a surjective morphism of left
quasigroups $f:Q\longrightarrow Q^{\prime}$, the relation
$\mathrm{ker}(f)=\\{(x,y)\,:\,f(x)=f(y)\\}$ is a congruence of $Q$ and
$Q^{\prime}\cong Q/\mathrm{ker}(f)$. Note also that if $\alpha$ is a
congruence of $Q$ and $[x]_{\alpha}\in E(Q/\alpha)$ then the block of $x$ with
respect to $\alpha$ is a subalgebra of $Q$.

Let $N\leq\mathrm{LMlt}(Q)$. We can define two equivalence relation out of
$N$, namely:

$\displaystyle x\,\mathcal{O}_{N}\,y\,$ $\displaystyle\text{if and only if
}x=h(y)\,\text{ for some }h\in N,$ $\displaystyle x\,\mathrm{con}_{N}\,y$
$\displaystyle\text{ if and only if }L_{x}L_{y}^{-1}\in N.$

The assignments above define two monotone operators
$\mathrm{con}_{*}:N\mapsto\mathrm{con}_{N}$ and
$\mathcal{O}_{*}:N\mapsto\mathcal{O}_{N}$ from the lattice of subgroups of
$\mathrm{LMlt}(Q)$ to $Equiv(Q)$.

If $\mathcal{O}_{N}=1_{Q}$ (i.e. $N$ is transitive on $Q$), we say that $Q$ is
connected by $N$. If $\mathrm{LMlt}(Q)$ is transitive we simply say that $Q$
is connected. If all the subalgebras of $Q$ are connected we say that $Q$ is
superconnected (in particular $Q$ is connected). The class of superconnected
left quasigroups has been studied in [BF21] and [Bon22b]. Connected idempotent
left quasigroups and left quasigroup with a Mal’cev term are connected by
their displacement group [BF21, Proposition 3.6].

Assume that $\alpha\in Con(Q)$, then we have a morphism of groups defined as:

(1) $\pi_{\alpha}:\mathrm{Dis}(Q)\longrightarrow\mathrm{Dis}(Q/\alpha),\quad
L_{x_{1}}^{k_{1}}\ldots L_{x_{n}}^{k_{n}}\longrightarrow
L_{[x_{1}]}^{k_{1}}\ldots L_{[x_{n}]}^{k_{n}}.$

Let us denote by $\mathrm{Dis}(Q)_{x}$ the pointwise stabilizer of $x$ in
$\mathrm{Dis}(Q)$. If $\mathrm{Dis}(Q)_{x}=1$ for every $x\in Q$ we say that
$Q$ is semiregular. In particular, we have

(2)
$\displaystyle\mathrm{Dis}_{\alpha}\leq\mathrm{Dis}^{\alpha}=\mathrm{ker}(\pi_{\alpha})\leq\mathrm{Dis}(Q)_{[x]}=\pi_{\alpha}^{-1}(\mathrm{Dis}(Q/\alpha)_{[x]})=\\{h\in\mathrm{Dis}(Q)\,:\,h(x)\,\alpha\,x\\}$

for every $x\in Q$. Moreover, we have the following.

######  Proposition 1.1.

[BS21, Proposition 3.2] Let $(Q,\cdot,\backslash)$ be a left quasigroup and
$\alpha,\beta\in Con(Q)$ such that $\alpha\leq\beta$. Then:

  * (i)

$\mathrm{Dis}_{\beta/\alpha}=\pi_{\alpha}(\mathrm{Dis}_{\beta})$.

  * (ii)

$\mathrm{Dis}^{\beta/\alpha}=\pi_{\alpha}(\mathrm{Dis}^{\beta}).$

We focus on the set of admissible subgroups contained in the displacement
group and its interplay with congruences. The admissible subgroups we are
interest in are defined in [Bon21] as

$\displaystyle\mathrm{Norm}^{\prime}(Q)$
$\displaystyle=\\{N\trianglelefteq\mathrm{LMlt}(Q)\,:\,N\leq\mathrm{Dis}(Q),\,\mathcal{O}_{N}\leq\mathrm{con}_{N}\\}$
$\displaystyle=\\{N\trianglelefteq\mathrm{LMlt}(Q)\,:\,\mathrm{Dis}_{\mathcal{O}_{N}}\leq
N\leq\mathrm{Dis}(Q)\\}.$

In particular
$\mathrm{Dis}_{\alpha},\mathrm{Dis}^{\alpha}\in\mathrm{Norm}^{\prime}(Q)$ for
every $\alpha\in Con(Q)$ [Bon21, Corollary 1.9]. If $N\in\mathrm{Norm}(Q)$
then $\mathcal{O}_{N}\in Con(Q)$ and $N\leq\mathrm{Dis}^{\mathcal{O}_{N}}$
[Bon21, Corollary 1.9]. It is easy to check that

$\mathcal{O}_{\mathrm{Dis}_{\alpha}}\leq\mathcal{O}_{\mathrm{Dis}^{\alpha}}\leq\alpha\leq\mathrm{con}_{\mathrm{Dis}_{\alpha}}\leq\mathrm{con}_{\mathrm{Dis}^{\alpha}}$

for every congruence $\alpha$. Given $N\in\mathrm{Norm}^{\prime}(Q)$, the
relation $\mathrm{con}_{N}$ does not need to be a congruence (this is the case
for semimedial and spelling left quasigroups including racks, see [Bon21,
Theorem 3.5, Theorem 5.6]).

The set of admissible subgroups is a sublattice of the lattice of normal
subgroups of the left multiplication group (see [Bon21, Lemma 1.7]), and is
stable under the usual correspondence between normal subgroups.

######  Corollary 1.2.

[Bon22a, Proposition 2.4] Let $(Q,\cdot,\backslash)$ be a left quasigroup and
$\alpha$ be a congruence of $Q$. Then the mappings

$\displaystyle\\{N\in\mathrm{Norm}^{\prime}(Q)\,:\,\mathrm{Dis}^{\alpha}\leq
N\\}$ $\displaystyle\longleftrightarrow\mathrm{Norm}^{\prime}(Q/\alpha)$
$\displaystyle N$ $\displaystyle\mapsto\pi_{\alpha}(N)$
$\displaystyle\pi_{\alpha}^{-1}(K)$ $\displaystyle\leftmapsto K$

provides an isomorphism of lattices.

The following lemma shows the interplay between the maps $\pi_{\alpha}$ as
defined in (1) and the operators $\mathcal{O}_{*}$ and $\mathrm{con}_{*}$.

######  Lemma 1.3.

Let $(Q,\cdot,\backslash)$ be a left quasigroup, $\alpha\in Con(Q)$,
$\beta=\mathcal{O}_{\mathrm{Dis}^{\alpha}}$ and
$N\in\mathrm{Norm}^{\prime}(Q)$. If $\mathrm{Dis}^{\alpha}\leq N$ then:

  * (i)

$\mathcal{O}_{N}/\beta=\mathcal{O}_{\pi_{\beta}(N)}$.

  * (ii)

$\mathrm{con}_{N}/\alpha=\mathrm{con}_{\pi_{\alpha}(N)}$.

###### Proof.

Since $\mathrm{Dis}^{\alpha}\leq N$ we have that $\beta\leq\mathcal{O}_{N}$
and $\alpha\leq\mathrm{con}_{\mathrm{Dis}^{\alpha}}\leq\mathrm{con}_{N}$.

(i) By definition $[x]_{\beta}\,\mathcal{O}_{N}/\beta\,[y]_{\beta}$ if and
only if $x=h(y)$ for some $h\in N$. Thus, if $x=h(y)$ it follows that
$[x]_{\beta}=[h(y)]_{\beta}=\pi_{\beta}(h)([y]_{\beta})$ and so
$[x]_{\beta}\,\mathcal{O}_{N}/\beta\,[y]_{\beta}$. On the other hand if
$[x]_{\beta}=\pi_{\beta}(h)([y]_{\beta})=[h(y)]_{\beta}$ then
$x\,\beta\,h(y)$, i.e. $x=kh(y)$ for some $k\in\mathrm{Dis}^{\alpha}$. Since
$\mathrm{Dis}^{\alpha}\leq N$ we have that $hk\in N$ and thus
$x\,\mathcal{O}_{N}y$.

(ii) If $x\,\mathrm{con}_{N}\,y$ then
$\pi_{\alpha}(L_{x}L_{y}^{-1})=L_{[x]_{\alpha}}L_{[y]_{\alpha}}^{-1}\in\pi_{\alpha}(N)$.
On the other hand, if
$[x]_{\alpha}\,\mathrm{con}_{\pi_{\alpha}(N)}\,[y]_{\alpha}$ then
$L_{x}L_{y}^{-1}\in N\mathrm{Dis}^{\alpha}=N$. ∎

###  1.3. The Cayley kernel

The Cayley kernel of a left quasigroup $Q$ is the equivalence relation
$\lambda_{Q}=\mathrm{con}_{1}$, i.e. given $x,y\in Q$ then

$x\,\lambda_{Q}\,y\,\text{ if and only if }L_{x}=L_{y}.$

In general, the Cayley kernel is not a congruence. If this is the case, we say
that $Q$ is a Cayley left quasigroup (e.g. racks are Cayley left quasigroups).

######  Remark 1.4.

[Bon22a, Remark 2.7] Let $(Q,\cdot,\backslash)$ be a left quasigroup and
$\alpha\in Con(Q)$:

  * (i)

$\alpha\leq\lambda_{Q}$ if and only if $\mathrm{Dis}_{\alpha}=1$;

  * (ii)

$\lambda_{Q/\alpha}=\mathrm{con}_{\mathrm{Dis}^{\alpha}}/\alpha$.

In [BS19] we show that strongly abelian congruences of left quasigroups in the
sense of [HD88] are those below the Cayley kernel. In particular, congruence
arising from orbits are related to strongly abelian congruences.

######  Lemma 1.5.

[Bon22a, Lemma 2.9] Let $(Q,\cdot,\backslash)$ be a left quasigroup,
$\alpha\in Con(Q)$, $\beta=\mathcal{O}_{\mathrm{Dis}^{\alpha}}$ and
$\gamma=\mathcal{O}_{\mathrm{Dis}_{\alpha}}$. Then
$\alpha/\beta\leq\lambda_{Q/\beta}$ and $\alpha/\gamma\leq\lambda_{Q/\gamma}$.

A left quasigroup $Q$ is said to be sharp  if given $\alpha,\beta\in Con(Q)$,
$\alpha\leq\beta$ and $\beta/\alpha\leq\lambda_{Q/\alpha}$ then
$\alpha=\beta$. The property of being sharp has also an universal algebraic
interpretation: indeed a left quasigroup $Q$ is sharp if and only if every
homomorphic image of $Q$ omit strongly abelian congruences.

######  Corollary 1.6.

The class of sharp left quasigroups is closed under homorphic images.

Let us show a characterization of sharp left quasigroups in terms of the
operators $\mathcal{O}_{*}$, $\mathrm{Dis}_{*}$ and $\mathrm{Dis}^{*}$.

######  Lemma 1.7.

Let $(Q,\cdot,\backslash)$ be a left quasigroup. The following are equivalent:

  * (i)

$Q$ is sharp .

  * (ii)

$\mathcal{O}_{\mathrm{Dis}_{\alpha}}=\mathcal{O}_{\mathrm{Dis}^{\alpha}}=\alpha$
for every $\alpha\in Con(Q)$.

###### Proof.

(i) $\Rightarrow$ (ii) According to Lemma 1.5 We have
$\gamma=\mathcal{O}_{\mathrm{Dis}_{\alpha}}\leq\beta=\mathcal{O}_{\mathrm{Dis}^{\alpha}}\leq\alpha$
and $\alpha/\gamma\leq\lambda_{Q/\gamma}$ for every $\alpha\in Con(Q)$. Hence,
if $Q$ is sharp $\gamma=\beta=\alpha$.

(ii) $\Rightarrow$ (i) Let $\alpha,\beta\in Con(Q)$. Assume that
$\beta\leq\alpha$ and $\alpha/\beta\leq\lambda_{Q/\beta}$. Then
$\mathrm{Dis}_{\alpha/\beta}=1$, i.e.
$\mathrm{Dis}_{\alpha}\leq\mathrm{Dis}^{\beta}$. So we have that

$\alpha=\mathcal{O}_{\mathrm{Dis}_{\alpha}}\leq\mathcal{O}_{\mathrm{Dis}^{\beta}}=\beta$

and so $\alpha=\beta$. ∎

It is immediate to see that if $Q$ is a sharp left quasigroup, then
$\mathcal{O}_{*}$ is onto and $\mathrm{Dis}^{*}$ and $\mathrm{Dis}_{*}$ are
injective on $Q$.

A left quasigroup $Q$ is faithful if $\lambda_{Q}=0_{Q}$ (note that faithful
left quasigroups are Cayley).

######  Lemma 1.8.

Let $(Q,\cdot,\backslash)$ be a left quasigroup. The following are equivalent:

  * (i)

$\mathcal{O}_{*}=\mathrm{con}_{*}$ on $Q$.

  * (ii)

$\mathcal{O}_{\mathrm{Dis}_{\alpha}}=\mathcal{O}_{\mathrm{Dis}^{\alpha}}=\alpha=\mathrm{con}_{\mathrm{Dis}_{\alpha}}=\mathrm{con}_{\mathrm{Dis}^{\alpha}}$
for every $\alpha\in Con(Q)$.

  * (iii)

$Q/\alpha$ is faithful for every $\alpha\in Con(Q)$.

In particular, if (i) holds then $Q$ is sharp .

###### Proof.

(i) $\Rightarrow$ (ii) Let $\alpha\in Con(Q)$. Since we have

$\mathcal{O}_{\mathrm{Dis}_{\alpha}}\leq\mathcal{O}_{\mathrm{Dis}^{\alpha}}\leq\alpha\leq\mathrm{con}_{\mathrm{Dis}_{\alpha}}\leq\mathrm{con}_{\mathrm{Dis}^{\alpha}}$

if $\mathcal{O}_{*}=\mathrm{con}_{*}$ equality holds.

(ii) $\Rightarrow$ (iii) According to Remark 1.4(ii)
$\lambda_{Q/\alpha}=\mathrm{con}_{\mathrm{Dis}^{\alpha}}/\alpha$. So we have
that $\lambda_{Q/\alpha}=0_{Q}$.

(iii) $\Rightarrow$ (i) Let $N\in\mathrm{Norm}(Q)$ and let
$\alpha=\mathcal{O}_{N}\leq\mathrm{con}_{N}$. Thus,
$\lambda_{Q/\alpha}=\mathrm{con}_{\mathrm{Dis}^{\alpha}}/\alpha=0_{Q/\alpha}$
i.e. $\mathrm{con}_{\mathrm{Dis}^{\alpha}}=\alpha$. Since
$N\leq\mathrm{Dis}^{\alpha}$ then
$\mathrm{con}_{N}\leq\mathrm{con}_{\mathrm{Dis}^{\alpha}}=\alpha$, and so
equality holds. ∎

The property in Lemma 1.8(iii) is clearly closed under homomorphic images.

##  2\. Galois connections for left quasigroups

Let $(A,\leq)$ and $(B,\leq)$ be two posets. A monotone Galois connection
between these posets consists of two monotone functions: $F:A\longrightarrow
B$ and $G:B\longrightarrow A$, such that for all $x\in A$ and $y\in B$, we
have

$F(x)\leq y\,\text{ if and only if }\,x\leq G(y).$

If $(F,G)$ is a Galois connection between two posets, then $FGF=F$ and
$GFG=G$. In particular, if $F$ (resp. $G$) is an isomorphism then $F^{-1}=G$.
In such case we say that the pair $(F,G)$ provides an isomorphism between the
two posets.

The following Galois connection for left quasigroups was first introduced for
racks in [BS21] and then for left quasigroups in general in [Bon21].

######  Theorem 2.1.

[Bon21, Theorem 1.10] Let $(Q,\cdot,\backslash)$ be a left quasigroup. The
pair of mappings $\mathcal{O}_{*}$ and $\mathrm{Dis}^{*}$ provides a monotone
Galois connection between $Con(Q)$ and $\mathrm{Norm}^{\prime}(Q)$.

Let $(Q,\cdot,\backslash)$ be a left quasigroup and $\alpha\in Con(Q)$.
According to Lemma 1.1, Corollary 1.2 and Theorem 2.1 the following diagram
(where the vertical arrow labeled by $*/\alpha$ is the canonical isomorphism
$\beta\mapsto\beta/\alpha$) is commutative:

(3) $\textstyle{\\{\beta\in
Con(Q)\,:\,\alpha\leq\beta\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Dis}^{*}}$$\scriptstyle{*/\alpha}$$\textstyle{\\{N\in\mathrm{Norm}^{\prime}(Q)\,:\,\mathrm{Dis}^{\alpha}\leq
N\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\alpha}}$$\textstyle{Con(Q/\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Dis}^{*}}$$\textstyle{\mathrm{Norm}^{\prime}(Q/\alpha)}$

Let $\beta=\mathcal{O}_{\mathrm{Dis}^{\alpha}}$. The pair
$(\mathrm{Dis}^{*},\mathcal{O}_{*})$ provides a Galois connection and so
$\mathrm{Dis}^{\alpha}=\mathrm{Dis}^{\beta}$. According to Lemma 1.3(i) we
have the following commuting diagram:

(4) $\textstyle{\\{N\in\mathrm{Norm}^{\prime}(Q)\,:\,\mathrm{Dis}^{\beta}\leq
N\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{O}_{*}}$$\scriptstyle{\pi_{\beta}}$$\textstyle{\\{\gamma\in
Con(Q)\,:\,\beta\leq\gamma\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{*/\beta}$$\textstyle{\mathrm{Norm}^{\prime}(Q/\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{O}_{*}}$$\textstyle{Con(Q/\beta)}$

We say that $Q$ has the congruence determined by orbits (shortly, CDOs) if the
pair $(\mathrm{Dis}^{*},\mathcal{O}_{*})$ provides an isomophism between
$Con(Q)$ and $\mathrm{Norm}(Q)$. Note that if $Q$ has CDOs then
$1_{Q}=\mathcal{O}_{\mathrm{Dis}(Q)}$, namely $Q$ is connected by
$\mathrm{Dis}(Q)$.

By Corollary 1.2, the map $\pi_{\alpha}$ is bijective between
$\\{N\in\mathrm{Norm}^{\prime}(Q)\,:\,\mathrm{Dis}^{\alpha}\leq N\\}$ and
$\mathrm{Norm}^{\prime}(Q/\alpha)$. Then according to diagram (3), if
$\mathrm{Dis}^{*}$ is injective (resp. onto) on $Q$ then $\mathrm{Dis}^{*}$ is
also injective (resp. onto) on $Q/\alpha$.

######  Corollary 2.2.

The class of left quasigroups having CDOs is closed under homomorphic images.

The following example shows that having CDOs is a relevant property.

######  Example 2.3.

Let $Q$ be a simple left quasigroup, $N\in\mathrm{Norm}^{\prime}(Q)$ and
$\alpha=\mathcal{O}_{N}$. So either $\alpha=0_{Q}$ and so $N=1$, or
$\alpha=1_{Q}$ and thus $\mathrm{Dis}(Q)=\mathrm{Dis}_{\alpha}\leq N$.
Therefore, $\mathrm{Norm}^{\prime}(Q)=\\{1,\mathrm{Dis}(Q)\\}$. Note that
$\mathrm{Dis}(Q)=1$ if and only if $\lambda_{Q}=1_{Q}$. We have to discuss two
cases:

  * (i)

Let $\lambda_{Q}=1_{Q}$. If $Q$ is idempotent then $Q\cong\mathcal{P}_{2}$,
otherwise $Q\cong\mathrm{Aff}(\mathbb{Z}_{p},0,1,1)$ for $p$ prime.

  * (ii)

If $\lambda_{Q}\neq 1_{Q}$, we have
$\mathrm{Norm}^{\prime}(Q)=\\{1,\mathrm{Dis}(Q)\\}$ and accordingly $Q$ has
CDOs.

In particular, simple idempotent left quasigroups with size bigger than $2$
have CDOs.

Let us introduce a second Galois connection for left quasigroups.

######  Proposition 2.4.

Let $(Q,\cdot,\backslash)$ be a left quasigroup. The pair of mappings
$\mathrm{con}_{*}$ and $\mathrm{Dis}_{*}$ provides a monotone Galois
connection between $Equiv(Q)$ and the lattice of normal subgroups of
$\mathrm{LMlt}(Q)$.

###### Proof.

If $\alpha\leq\mathrm{con}_{N}$, then
$\\{L_{x}L_{y}^{-1}\,:\,x,\,\alpha\,y\\}\subseteq N$ and therefore
$\mathrm{Dis}_{\alpha}\leq N$. On the other hand, if
$\mathrm{Dis}_{\alpha}\leq N$, then
$\alpha\leq\mathrm{con}_{\mathrm{Dis}_{\alpha}}\leq\mathrm{con}_{N}$. ∎

Let $(Q,\cdot,\backslash)$ be a left quasigroup and $\alpha\in Con(Q)$.
According to Lemma 1.1, Corollary 1.2 and Theorem 2.1 the following diagram is
commutative:

(5) $\textstyle{\\{\beta\in
Con(Q)\,:\,\alpha\leq\beta\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Dis}_{*}}$$\scriptstyle{*/\alpha}$$\textstyle{\\{N\in\mathrm{Norm}^{\prime}(Q)\,:\,\mathrm{Dis}_{\alpha}\leq
N\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\alpha}}$$\textstyle{Con(Q/\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Dis}_{*}}$$\textstyle{\mathrm{Norm}^{\prime}(Q/\alpha)}$

Let $(Q,\cdot,\backslash)$ be a left quasigroup and $\alpha\in Con(Q)$. The
following diagram is commutative by Lemma 1.3(ii):

(6) $\textstyle{\\{N\in\mathrm{Norm}^{\prime}(Q)\,:\,\mathrm{Dis}^{\alpha}\leq
N\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{con}_{*}}$$\scriptstyle{\pi_{\alpha}}$$\textstyle{\\{\beta\in
Equiv(Q)\,:\,\alpha\leq\beta\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{*/\alpha}$$\textstyle{\mathrm{Norm}^{\prime}(Q/\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{con}_{*}}$$\textstyle{Equiv(Q/\alpha)}$

We say that $Q$ has the congruence determined by subgroups (shortly, CDSg) if
the pair $(\mathrm{Dis}_{*},\mathrm{con}_{*})$ provides an isomophism between
$Con(Q)$ and $\mathrm{Norm}^{\prime}(Q)$.

The map $\pi_{\alpha}$ is bijective between
$\\{N\in\mathrm{Norm}^{\prime}(Q)\,:\,\mathrm{Dis}^{\alpha}\leq N\\}$ and
$\mathrm{Norm}^{\prime}(Q/\alpha)$. Then according to diagram (6), if
$\mathrm{con}_{*}$ is injective (resp. onto) on $Q$ then $\mathrm{con}_{*}$ is
also injective (resp. onto) on $Q/\alpha$.

######  Corollary 2.5.

The class of left quasigroups having CDSg is closed under homomorphic images.

######  Example 2.6.

Let us show some examples of left quasigroups having CDSg.

  * (i)

Let $Q$ be a simple Cayley left quasigroup. If $\lambda_{Q}=1_{Q}$ then either
$Q=\mathcal{P}_{2}$ or $Q=\mathrm{Aff}(\mathbb{Z}_{p},0,1,1)$ for $p$ prime.
Otherwise $\lambda_{Q}=0_{Q}$ and $Q$ has CDSg.

  * (ii)

Superconnected idempotent semiregular left quasigroup are quandles (see
[Bon22a, Lemma 3.3]). Such quandles have CDSg [Bon22b, Proposition 2.12].

The following proposition extends [BS21, Proposition 3.13].

######  Theorem 2.7.

Let $(Q,\cdot,\backslash)$ be a left quasigroup. The following are equivalent:

  * (i)

$Q$ has CDSg.

  * (ii)

$Q$ is sharp and has CDOs.

  * (iii)

$\mathrm{con}_{*}=\mathcal{O}_{*}$ and $\mathrm{Dis}_{*}=\mathrm{Dis}^{*}$ on
$Q$.

###### Proof.

(i) $\Rightarrow$ (ii) Clearly
$\mathrm{Dis}_{\lambda_{Q}}=\mathrm{Dis}_{\mathrm{con}_{1}}=1=\mathrm{Dis}_{0_{Q}}$
and so $\lambda_{Q}=0_{Q}$. Therefore $Q$ and its factors are all faithful. By
Lemma 1.8 $Q$ is sharp and we have that $\mathrm{con}_{*}=\mathcal{O}_{*}$.
Since $\mathrm{con}_{*}$ is an isomorphism between $\mathrm{Norm}^{\prime}(Q)$
and $Con(Q)$ then so it is $\mathcal{O}_{*}$.

(ii) $\Rightarrow$ (iii) By Lemma 1.7 we have that
$\mathcal{O}_{\mathrm{Dis}_{\alpha}}=\mathcal{O}_{\mathrm{Dis}^{\alpha}}$ for
every $\alpha\in Con(Q)$. Since $Q$ has CDOs we have that
$\mathrm{Dis}_{*}=\mathrm{Dis}^{*}$ on $Q$ and accordingly
$\mathcal{O}_{*}=(\mathrm{Dis}^{*})^{-1}=(\mathrm{Dis}_{*})^{-1}=\mathrm{con}_{*}$
on $Q$.

(iii) $\Rightarrow$ (i) Let $N\in\mathrm{Norm}^{\prime}(Q)$ and
$\alpha=\mathcal{O}_{N}=\mathrm{con}_{N}$. Then $\mathrm{Dis}_{\alpha}\leq
N\leq\mathrm{Dis}^{\alpha}$ and so $N=\mathrm{Dis}_{\mathrm{con}_{N}}$. On the
other hand, according to Lemma 1.8 we have that
$\alpha=\mathrm{con}_{\mathrm{Dis}_{\alpha}}$ for every $\alpha\in Con(Q)$. ∎

Some classes of left quasigroups are know to be sharp :

  * (i)

Malt’sev left quasigroups (indeed they omit strongly abelian congruences
[HD88]).

  * (ii)

Superconnected idempotent left quasigroups (indeed they are faithful and so
they are all their factors [Bon22b]).

######  Corollary 2.8.

Let $Q$ be a Malt’sev (resp. idempotent superconnected) left quasigroup. The
following are equivalent:

  * (i)

$Q$ has CDOs.

  * (ii)

$\mathcal{Q}$ has CDSg.

The following example shows that having CDOs and having CDSg are dinstinct
properties in general.

######  Example 2.9.

The quandle $Q=$SmallQuandle(8,1) in the RIG database of GAP [Ven15] has CDOs,
but it has not CDSg (indeed $Q$ is not sharp ).

##  3\. Nilpotency and CDSg property

###  3.1. Commutator theory and central extensions

We recall the basics of commutator theory for the reader’s convenience. The
commutator theory for arbitrary algebraic structures have been developed in
[H.76, FM87], including the definition of commutator of congruences and the
related notions of center, solvability and nilpotency. We will present all the
definitions just for left quasigroups.

Let $(Q,\cdot,\backslash)$ be a left quasigroup and $\alpha,\beta,\delta\in
Con(Q)$. We say that _$\alpha$ centralizes $\beta$ over $\delta$_, and write
$C(\alpha,\beta;\delta)$, if for every $(n+1)$-ary term operation $t$, every
pair $x\,\alpha\,y$ and every $z_{1}\,\beta\,u_{1}$, $\dots$,
$z_{n}\,\beta\,u_{n}$ we have

$t^{Q}(x,z_{1},\dots,z_{n})\,\delta\,t^{Q}(x,u_{1},\dots,u_{n})\quad\text{implies}\quad
t^{Q}(y,z_{1},\dots,z_{n})\,\delta\,t^{Q}(y,u_{1},\dots,u_{n}).$

We denote by $[\alpha,\beta]$ the _commutator_ of $\alpha,\beta\in Con(Q)$,
that is the smallest congruence $\delta$ such that $C(\alpha,\beta;\delta)$. A
congruence $\alpha$ is called:

  * •

_abelian_ if $C(\alpha,\alpha;0_{Q})$, i.e., if $[\alpha,\alpha]=0_{Q}$,

  * •

_central_ if $C(\alpha,1_{Q};0_{Q})$, i.e., if $[\alpha,1_{Q}]=0_{Q}$.

The largest congruence of $Q$ that centralizes $1_{Q}$ is called the _center_
of $Q$ and denoted by $\zeta_{Q}$. A left quasigroup $Q$ is called _abelian_
if $\zeta_{Q}=1_{Q}$, or, equivalently, if the congruence $1_{Q}$ is abelian.
We can define a series of congruence of $Q$ that plays the same role as the
series of centers for groups. The definition is the following:

$\zeta_{1}(Q)=\zeta_{Q},\qquad\zeta_{n+1}(Q)/\zeta_{n}(Q)=\zeta_{(Q/\zeta_{n}(Q))}$

for every $n\in\mathbb{N}$. The left quasigroup $Q$ is called _nilpotent_ of
length $n$ if $\zeta_{n}(Q)=1_{Q}$.

The theory of commutators have been adapted to the setting of racks and
quandles in [BS21]. The main results can also be partially applied to the
setting of left quasigroups.

######  Lemma 3.1.

[BS21, Lemma 5.1] Let $(Q,\cdot,\backslash)$ be a left quasigroup,
$\alpha,\beta\in Con(Q)$. If $C(\alpha,\beta;0_{Q})$ holds then
$[\mathrm{Dis}_{\alpha},\mathrm{Dis}_{\beta}]=1$ and
$\left(\mathrm{Dis}_{\beta}\right)_{x}=\left(\mathrm{Dis}_{\beta}\right)_{y}$
whenever $x\,\alpha\,y$.

######  Corollary 3.2.

Let $(Q,\cdot,\backslash)$ be a left quasigroup and $\alpha\leq\zeta_{Q}$.
Then $\mathrm{Dis}_{\alpha}$ is central in $\mathrm{Dis}(Q)$ and
$\mathrm{Dis}(Q)_{x}=\mathrm{Dis}(Q)_{y}$ whenever $x\,\alpha\,y$.

Let us present a standard contruction of left quasigroups involving abelian
groups. Later in the section we are proving that such construction is related
to central congruences. Let $(Q,\cdot,\backslash)$ be a left quasigroup, $A$
an abelian group, $f\in\mathrm{Aut}(A)$, $g\in\mathrm{End}(A)$ and
$\theta:Q\times Q\longrightarrow A$ be a map. We can define the left
quasigroup $E=(Q\times A,\cdot,\backslash)$ where

(7) $(x,a)\cdot(y,b)=(x\cdot y,g(a)+f(b)+\theta(x,y)).$

We denote such a structure by $\mathrm{Aff}(Q,A,g,f,\theta)$ and we say that
$E$ is a central extension of $Q$ by $A$.

Moreover, the map

$p_{1}:E\longrightarrow Q,\quad(x,a)\mapsto x$

is a morphism of left quasigroups.

If $|Q|=1$, we can identify $Q\times A$ with $A$ and (7) reads

$a\cdot b=g(a)+f(b)+c$

for some $c\in A$. For this special case we use the notation
$E=\mathrm{Aff}(A,g,f,c)$ and we say that $E$ is an affine left quasigroup
over $A$.

######  Lemma 3.3.

Let $(Q,\cdot,\backslash)$ be a left quasigroup, $A$ an abelian group and
$E=\mathrm{Aff}(Q,A,g,f,\theta)$. Then every term operations of $E$ has the
form

$t^{E}((x_{1},a_{1}),\ldots,(x_{n},a_{n}))=(t^{Q}(x_{1},\ldots
x_{n}),\sum_{j=1}^{n}G_{j}(a_{j})+\Theta(x_{1},\ldots x_{n}))$

where $G_{j}\in\mathrm{End}(A)$ for every $j=1,\ldots n$ and
$\Theta:Q^{n}\longrightarrow A$.

###### Proof.

Let us prove the statement by induction on the number of occurrences of
variables. If the term $t$ is just a variable there is nothing to prove. Let
us denote $\overline{x}=(x_{1},\ldots,x_{n})$ and
$\overline{(x,a)}=((x_{1},a_{1}),\ldots,(x_{n},a_{n}))$. Then

$t^{E}(\overline{(x,a)})=L_{s_{1}^{E}(\overline{(x,a)})}s_{2}^{E}(\overline{(x,a)})$

for suitable subterms $s_{1}$ and $s_{2}$. Then by induction we have that

$s_{i}^{E}(\overline{(x,a)})=(s_{i}^{Q}(\overline{x}),\sum_{j}G_{i,j}(a_{j})+\Theta_{i}(\overline{x}))$

for $i=1,2$. Therefore

$\displaystyle L_{s_{1}^{E}(\overline{(x,a)})}s_{2}^{E}(\overline{(x,a)})$
$\displaystyle=(L_{s_{1}^{Q}(\overline{x})}s_{2}^{Q}(\overline{x})),g\left(\sum_{j}G_{1,j}(a_{j})+\Theta_{1}(\overline{x})\right)+f\left(\sum_{j}G_{2,j}(a_{j})+\Theta_{2}(\overline{x})\right)+\theta(s_{1}^{Q}(\overline{x}),s_{2}^{Q}(\overline{x})))$
$\displaystyle=(t^{Q}(\overline{x}),\sum_{j}\underbrace{(gG_{1,j}+fG_{2,j})}_{\in\mathrm{End}(A)}(a_{j})+\underbrace{g(\Theta_{1}(\overline{x}))+f(\Theta_{2}(\overline{x}))+\theta(s_{1}^{Q}(\overline{x}),s_{2}^{Q}(\overline{x}))}_{=\Theta^{\prime}(\overline{x})})$

and so the statement follows. ∎

######  Corollary 3.4.

Let $(Q,\cdot,\backslash)$ be a left quasigroup and
$E=\mathrm{Aff}(Q,A,g,f,\theta)$. Then $\mathrm{ker}(p_{1})$ is a central
congruence of $E$.

###### Proof.

According to Lemma 3.3, we have

$t^{E}((x,a),(y_{1},b_{1})\ldots,(y_{n-1},b_{n-1}))=(t^{Q}(x,y_{1},\ldots,y_{n-1}),G(a)+\sum_{j}G_{j}(b_{j})+\Theta(x,y_{1},\ldots
y_{n-1}))$

where $G,G_{j}\in\mathrm{End}(A)$ and $\Theta:Q^{n}\longrightarrow A$. If the
equality

$t^{E}((x,a),(y_{1},b_{1})\ldots,(y_{n-1},b_{n-1}))=t^{E}((x,a),(z_{1},c_{1})\ldots,(z_{n-1},c_{n-1}))$

holds then

$\displaystyle t^{Q}(x,y_{1},\ldots,y_{n-1})$
$\displaystyle=t^{Q}(x,z_{1},\ldots,z_{n-1}),$
$\displaystyle\sum_{j}G_{j}(b_{j})+\Theta(x,y_{1},\ldots,y_{n-1})$
$\displaystyle=\sum_{j}G_{j}(c_{j})+\Theta(x,z_{1},\ldots,z_{n-1}).$

Hence we also have

$\displaystyle t^{E}((x,d),(y_{1},b_{1})\ldots(y_{n-1},b_{n-1}))$
$\displaystyle=(t^{Q}(x,y_{1},\ldots,y_{n-1}),G(d)+\sum_{j}G_{j}(b_{j})+\Theta(x,y_{1},\ldots
y_{n-1}))$
$\displaystyle=(t^{Q}(x,z_{1},\ldots,z_{n-1}),G(d)+\sum_{j}G_{j}(c_{j})+\Theta(x,z_{1},\ldots
z_{n-1}))$ $\displaystyle=t^{E}((x,d),(z_{1},c_{1}),\ldots(z_{n-1},c_{n-1}))$

for every $d\in A$. ∎

According to Corollary 3.2 $\mathrm{Dis}_{\mathrm{ker}(p_{1})}\leq
Z(\mathrm{Dis}(E))$ since $\mathrm{ker}(p_{1})$ is a central congruence by
Corollary 3.4. Thus, if $h=wL_{(z,d)}^{k}$ for $w\in\mathrm{Dis}(Q)$ and
$k\in\mathbb{Z}$ we have that the action of the generators of
$\mathrm{Dis}_{\mathrm{ker}(p_{1})}$ is given by

(8) $\displaystyle
hL_{(x,a)}L_{(x,b)}^{-1}h^{-1}(y,c)=L_{(z,d)}^{k}L_{(x,a)}L_{(x,b)}^{-1}L_{(z,d)}^{-k}(y,c)=(y,c+f^{k}g(a-b))$

for every $x,y\in Q$ and every $a,b,c\in A$.

Let $N\leq A$. We define the relation $\alpha_{N}$ by setting

(9) $(x,a)\,\alpha_{N}\,(y,b)\,\text{ if and only if }\,x=y\,\text{ and
}\,a-b\in N$

for every $(x,a),(y,b)\in Q\times A$.

######  Lemma 3.5.

Let $(Q,\cdot,\backslash)$ be a left quasigroup, $A$ an abelian group,
$E=\mathrm{Aff}(Q,A,g,f,\theta)$ and $N\leq A$. If $g(N)\leq N=f(N)$, the
relation $\alpha_{N}$ is a congruence of $E$.

###### Proof.

Let $a-b\in N$. Then, $f^{k}g(a-b)\in N$ for every $k\in\mathbb{Z}$ and so
according to (8) we have

$hL_{(x,a)}L_{(x,b)}^{-1}h^{-1}(y,c)=(y,c+f^{k}g(a-b))\,\alpha_{N}\,(y,c)$

for every $(y,c)\in E$. Therefore
$\mathrm{Dis}_{\alpha_{N}}\leq\mathrm{Dis}^{\alpha_{N}}$. Moreover, using (7)

$\displaystyle(y,c)(x,a)$
$\displaystyle=(yx,g(c)+f(a)+\theta_{y,x})\,\alpha_{N}\,(y,c)(x,b)=(yx,g(c)+f(b)+\theta_{y,x}),$
$\displaystyle(y,c)\backslash(x,a)$ $\displaystyle=(y\backslash
x,f^{-1}(a-g(c)-\theta_{y,y\backslash
x}))\,\alpha_{N}\,(y,c)\backslash(x,b)=(y\backslash
x,f^{-1}(b-g(c)-\theta_{y,y\backslash x})).$

Therefore $\alpha_{N}$ provides a system of blocks for the action of
$\mathrm{LMlt}(E)$ and so $\alpha_{N}$ is a congruence according to [Bon21,
Lemma 1.5]. ∎

Let us show that the converse of [Bon22a, Lemma 2.3] holds for congruences
arising from central extensions.

######  Lemma 3.6.

Let $Q$ be an idempotent left quasigroup, $E=\mathrm{Aff}(Q,A,g,f,\theta)$ and
$\alpha=\mathrm{ker}(p_{1})$. If $\alpha=\mathcal{O}_{\mathrm{Dis}_{\alpha}}$
then the blocks of $\alpha$ are connected.

###### Proof.

The blocks of $\alpha$ are subalgebras of $E$, since $Q$ is idempotent. The
group $\mathrm{Dis}_{\alpha}$ is generated by
$\\{hL_{(x,a)}L_{(x,b)}^{-1}h^{-1}\,:\,x\in Q,\,a,b\in A\\}$.

According to (8) the orbits of $(y,c)$ with respect to the action of
$\mathrm{Dis}_{\alpha}$ is $(y,c+H)$ where $H$ is the subgroup generated by
$\\{f^{k}g(A)\,:\,k\in\mathbb{Z}\\}$ and it coincides with the orbit with
respect to the action of $\mathrm{Dis}([x]_{\alpha})$ (indeed it is enough to
set $x=y=z$ in (8)). ∎

###  3.2. Nilpotent left quasigroups having CDSg

Note that $E=\mathrm{Aff}(Q,A,g,f,\theta)$ is idempotent if and only if $Q$ is
idempotent, $g=1-f$ and $\theta_{x,x}=0$. Note that in this case the blocks of
$\mathrm{ker}(p_{1})$ are affine quandles isomorphic to
$\mathrm{Aff}(A,1-f,f,0)$.

In this subsection we strengthen [Bon20, Proposition 3.17], since we extend it
to idempotent left quasigroups of arbitrary cardinality.

######  Lemma 3.7.

Let $E=\mathrm{Aff}(Q,A,1-f,f,\theta)$ be an idempotent left quasigroup. If
$E$ has CDOs then $\mathrm{Aff}(A,1-f,f,0)$ is superconnected.

###### Proof.

Let $a\in A$. The mapping $t_{a}:x\mapsto x+a$ is an automorphism of
$Q^{\prime}=\mathrm{Aff}(A,1-f,f,0)$. So if $M$ is a subquandle of
$Q^{\prime}$, then $M=a+M^{\prime}\cong M^{\prime}$ for some subquandle
$M^{\prime}$ containing $0$ and some $a\in A$. So, it is enough to show that
the subquandles containing $0$ of $Q^{\prime}$ are connected. Let $M$ be a
subquandle of $Q^{\prime}$ containing $0$. Let us denote by $N=\langle m,m\in
M\rangle\leq A$. Note that $f(m)=0*m\in M$ and $f^{-1}(m)=0\backslash m\in M$
for every $m\in M$. So $(1-f)(N)\leq N=f(N)$. In particular $N$ is a
subquandle of $Q^{\prime}$ and it contains $M$.

According to Lemma 3.5(ii), the relation $(x,a)\,\alpha_{N}\,(y,b)$ if and
only if $x=y$ and $a-b\in N$ is a congruence of $E$. The quandle $E$ has CDOs,
i.e. $\alpha_{N}=\mathcal{O}_{\mathrm{Dis}_{\alpha_{N}}}$ and by Lemma 3.6 the
blocks of $\alpha_{N}$ are connected. In particular $N=[(x,0)]_{\alpha_{N}}$
is connected. Moreover

$\mathrm{Dis}(N)=(1-f)(N)=(1-f)(\langle m,\,m\in
N\rangle)=\langle(1-f)(m),\,m\in M\rangle=\mathrm{Dis}(M).$

Then we have

$M\subseteq N=(x,0)^{\mathrm{Dis}(N)}=(x,0)^{\mathrm{Dis}(M)}\subseteq M,$

and so $N=M$ and $M$ is connected. ∎

In particular, Lemma 3.7 implies that affine quandles having CDOs are
superconnected. In the following theorem we characterize nilpotent idempotent
left quasigroups having CDSg.

######  Theorem 3.8.

Let $Q$ be a nilpotent idempotent left quasigroup. The following are
equivalent:

  * (i)

$Q$ has CDSg.

  * (ii)

$Q$ is a semiregular superconnected latin quandle.

###### Proof.

(i) $\Rightarrow$ (ii) By Proposition 2.7 $Q$ has CDOs and
$\mathrm{Dis}_{*}=\mathrm{Dis}^{*}$ on $Q$. If $Q$ is abelian, then it is
semiregular and so it is a superconnected quandle by [Bon22a, Lemma 3.3] and
Lemma 3.7. Assume that $Q$ is nilpotent of length $n+1$. The factor
$Q/\zeta_{Q}$ has CDSg according to Lemma 2.5. By induction on the nilpotency
length, $Q/\zeta_{Q}$ is a semiregular superconnected quandle. So we have
$\mathrm{Dis}(Q)_{x}\leq\mathrm{Dis}(Q)_{[x]}=\mathrm{Dis}^{\zeta_{Q}}=\mathrm{Dis}_{\zeta_{Q}}\leq
Z(\mathrm{Dis}(Q))$ (the first equality follows by Lemma [Bon22a, Lemma 3.1]
and the relative displacement group of a central congruence is central by
Corollary 3.2). The left quasigroup $Q$ is connected by $\mathrm{Dis}(Q)$,
therefore $\mathrm{Dis}(Q)_{x}$ is normal and then trivial. So $Q$ is a
semiregular quandle by [Bon22a, Lemma 3.3].

In particular $Q$ is connected, faithful and
$\zeta_{Q}=\mathcal{O}_{\mathrm{Dis}_{\zeta_{Q}}}$, so $Q$ is a central
extension of $Q/\zeta_{Q}$ [BS21, Proposition 7.9] and
$[x]_{\zeta_{Q}}\cong\mathrm{Aff}(A,1-f,f,0)$. The blocks of $\zeta_{Q}$ are
superconnected by Lemma 3.7 and accordingly $Q$ is also superconnected (the
class of idempotent superconnected left quasigroups is closed under
extensions, see [Bon22b, Corollary 1.12]).

Finally, superconnected nilpotent quandles are latin [Bon22b, Theorem 2.15].

(ii) $\Rightarrow$ (i) It follows by Example 2.6. ∎

Note that also for idempotent nilpotent left quasigroups the property of
having CDOs and having CDSg are different: indeed the quandle in Example 2.9
is nilpotent.

## References

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# Study of silicon photomultipliers for the readout of a lead/scintillating-
fiber calorimeter

F. Alemanno  P. Bernardini  A. Corvaglia  G. De Matteis  L. Martina  A.
Miccoli  M. Panareo  M.P. Panetta  C. Pinto  A. Surdo11footnotetext:
Corresponding author.

###### Abstract

The KLOE electromagnetic calorimeter is expected to be reused in the Near
Detector complex of the DUNE experiment at Fermilab. The possible substitution
of traditional Photomultiplier Tubes (PMTs) with Silicon Photomultipliers
(SiPMs) in the refurbished calorimeter is the object of this investigation. A
block of the KLOE lead-scintillating fiber calorimeter has been equipped with
light guides and external trigger scintillators. The signals induced by cosmic
rays and environmental radioactivity have been collected by SiPM arrays on one
side of the calorimeter, and by conventional PMTs on the opposite side.
Efficiency, stability, and timing resolution of SiPMs have been studied and
compared with KLOE-PMTs performance. Conclusions about the convenience of
substituting PMTs with SiPMs are drawn.

##  1 Introduction

Silicon Photomultipliers (SiPMs) are solid-state photodetectors [1], which
consist of a high-density matrix of SPADs (Single-Photon Avalanche Diodes)
operating in Geiger mode as independent detectors. The number of fired
photodiodes is proportional to the number of collected photons. The SiPM
performance improved continuously in the last years and presently SiPMs are
widely used in physics instrumentation, ranging from accelerator to
astroparticle physics experiments. In particular, they have potentiality and
specific advantages for calorimetry applications [2]. First of all, the SiPM
photodetection range is compatible with the typical wavelength-shifted light
of the scintillating fibers ($\sim$ 460 nm), and unlike the PMTs, SiPMs are
insensitive to magnetic fields. Furthermore, since SiPMs operate at low
voltage, the high voltage power supply would not be necessary, with
convenience in compactness and cost.

The present study aims to evaluate the compatibility of SiPM readout of the
KLOE electromagnetic calorimeter [3] and the possible capability of SiPMs to
allow improvements in efficiency and timing resolution over standard PMT
readout. This check is requested because the KLOE calorimeter is going to be
refurbished as an element of SAND (System for on-Axis Neutrino Detection) in
the Near Detector complex [4] of the DUNE experiment [5].

##  2 Experimental setup

The experimental setup used in this study is shown in Fig. 1 (photo) and Fig.
2 (schematic view).

Figure 1: Picture of the experimental setup. The ECAL is in the center. The
PMTs with the light guides are visible on the left side while the SiPMs are
attached to the box containing the light guides on the right side. Figure 2:
Representation of the experimental setup front (left) and side (right) view,
not in scale. The lead-fibers stack (ECAL) is divided into 14 cells
represented by the central blue elements. The instrumented photosensors are
shown on both sides of each cell of the central column and indicated in red
for the PMTs and blue for the SiPM arrays, connected with the ECAL through
light guides (in yellow). The scintillators of the external trigger are placed
on the top and bottom of the setup and illustrated with green rectangles.

The principal element (ECAL) is a cut-out of the KLOE calorimeter [3], already
used in another test [6]. It is composed of a stack of thick grooved lead
foils, alternating with layers of scintillating fibers (blue-green, type
Pol.Hi.Tech-0046) with the peak emission wavelength
$\lambda_{\mathrm{peak}}\approx 460$ nm. The composite density results in 5
g/cm3.

The upper part of the ECAL volume is segmented into 12 cells (section
$4.4\times 4.4$ cm2, horizontal length 40.0 cm), while two larger cells
(section $6.6\times 6.6$ cm2) occupy the lower part. This results in a total
ECAL thickness of $\approx 15~{}X_{0}$ ($\approx 2.7~{}X_{0}$ for each smaller
cell), considering a particle vertically crossing the calorimeter from top to
bottom.

Lucite light guides (Fig. 3), similar to the KLOE ones, are glued at both
edges of the cells in the central column. In order to obtain a good light
collection while optimizing the coupling between ECAL and photodetectors, the
light guides combine a tapered mixing part with a quadrangular entrance and
circular exit terminating with a Winston cone (exit section $\approx 500$
mm2). On one side (left in Fig. 2) they are coupled to Hamamatsu-R5946 1.5’
photomultipliers [7], with a cathode properly fitting the circular base. On
the opposite side, the readout is performed using arrays of $8\times 8$ or
$4\times 4$ SiPMs (Hamamatsu S13361-3050 series [8]). Since the substitution
of the 4880 KLOE light guides is an inconvenient and complex operation, the
SiPMs were chosen in order to cover as much as possible of the present KLOE
light guide surface. It was evaluated that a custom SiPM shaped to fit the
circular guides is not strictly necessary for these measurements.
Consequently, additional coupling devices are adopted to associate the SiPM
arrays with the circular exit of the light guides to account for the different
SiPM surfaces and to test possible minor adjustments of the KLOE-like guides.
The coupling solutions are shown in Figs. 2 and 3, and described here:

  * -

SiPM 1, a lucite adapter is glued to fit the $8\times 8$ SiPM array (area 666
mm2),

  * -

SiPM 2, not implemented,

  * -

SiPM 3, the $4\times 4$ array is directly coupled to the light guide (covered
fraction $\sim$170/500),

  * -

SiPM 4, a lucite adapter is glued to fit the $4\times 4$ SiPM array (area 170
mm2).

Figure 3: Lucite light guides used to couple PMTs and SiPMs with the
calorimeter cells connecting the square section of the calorimeter bars with
the circular one of the photodetectors. Left: The long light guides are used
for all the PMTs and SiPMs, while the short ones act as adapters for SiPM 1
and SiPM 4, in order to couple different surfaces and allow the maximum light
collection. Right: design of light guides (azure), adapters (orange), and SiPM
arrays (magenta).

The light guides and the adapters are coupled with the calorimeter using the
Polytec EP 601-LV optical cement (refractive index 1.53), and with PMTs and
SiPMs through a silicone paste. Moreover, a system of adjustable springs
pushes gently the PMTs (according to the KLOE design) and the SiPM arrays
toward the light guides. A CAEN FERS-DT5202 electronic board with its readout
software is used to characterize the SiPM performance (Sec. 3) and to select
events for the efficiency measurement (Sec. 4). The latter measurement is
based on an external trigger for cosmic rays made by 4 plastic scintillator
bars read out by PMTs (green elements in Fig. 2), placed on top (A, B) and
bottom (C, D) of the experimental setup. A Teledyne Lecroy Waverunner 640Zi
oscilloscope is employed for both SiPM and PMT data taking, along with CAEN
and Lecroy modules, to measure the timing resolutions (Sec. 5).

###  2.1 SiPMs and PMTs

The SiPMs used in this test are the MPPC arrays of the Hamamatsu S13361-3050
series. Both the available configurations ($4\times 4$ and $8\times 8$
channels) were mounted but only the performance of 16-channels arrays (SiPM 3
and 4) are reported in this paper 222 The $8\times 8$ array and its readout
printed circuit are bulky and difficult to fit in the KLOE light guides
assembling setup.. This specific MPPC series is chosen in order to achieve the
maximum Photo-Detection Efficiency (PDEMAX) close to the peak wavelength of
the scintillating fibers (typically PDE${}_{MAX}=40\%$ at $\lambda=450$ nm,
ranging from $\sim$30% to $\sim$60% as a function of the overvoltage).
Furthermore, the SiPM arrays are characterized by a large effective
photosensitive area of $3\times 3$ mm2 for each channel, a 50 $\mu$m pixel
size which ensures a wide dynamic range of photon counting, and a gain of
$\approx 10^{6}$. Other features are summarized in [8]. The SiPMs readout is
performed with a desktop front-end unit, CAEN FERS-DT5202, housing Weeroc
Citiroc-1A chips, which hosts a SiPM power supply. As said beforehand, in this
test the SiPM arrays were chosen in order to maximize the coverage of the
light guide. Since it’s excluded to substitute the single PMT KLOE channel
with a larger number of readout channels, the SiPM array is considered as one
element in the following measurements. Moreover, this arrangement makes the
measurements analogous to that of the PMT. The SiPM performance are compared
with that of the Hamamatsu R5946 1.5’ mesh photomultiplier already used in the
KLOE calorimeter. The quantum efficiency of this PMT is 18% at $\lambda=450$
nm.

##  3 SiPM characterization and light yield

The known dependence of the SiPM performance on the temperature [9, 10]
required to work at stable conditions ($\sim 23^{\circ}$C). The SiPM
calibration is made in a self-trigger acquisition configuration requiring at
least one channel over the threshold. Initially, the SiPM channels are
calibrated individually by voltage (VBIAS) regulation, in order to get the
same fixed rate ($\approx 3.5$ kHz) for each element of the array. In this
case, the signal in the SiPMs is generated by dark counts, cosmic charged
particles and environmental radioactivity. Afterwards, in order to select
mainly charged particles crossing the calorimeter, the data are taken
requiring the self-trigger on the 16 channels of each SiPM along with the AND
of PMT 3 and PMT 4 (trigger rate $\approx$ 1.4 Hz). The result is shown in
Fig. 4 left, where the histogram is filled with the ADC counts taken from each
single channel of the SiPM 3. The same procedure is applied to obtain a
similar spectrum for the SiPM 4.

The calibration works properly for all channels of both SiPMs. Indeed, the
peaks are placed in the same position (same ADC value) for all of them and we
do not observe any broadening of the peaks when many channels are plotted
together. Moreover, the SiPM channels show linearity and stability up to a
15-photoelectrons (p.e.) signal, as can be seen in Fig. 4 right. From the ADC
values of the peaks, it is possible to infer the formula to convert the ADC
signal in number of p.e. expressed as follows:

$N_{p.e.}=1\ +\ \frac{ADC-ADC_{1}}{\Delta},$ (3.1)

where $ADC_{1}$ is the ADC value for one photon (first peak after the
pedestal) and $\Delta$ is the ADC-distance between adjacent peaks. This
formula has been applied successfully to fit the number of p.e. associated
with each peak and the ADC counts of the peak (Fig. 4, right).

Figure 4: Left: photoelectron spectrum of SiPM 3 (entries from each single
channel). The spectrum is truncated at 2000 counts and the first peak is the
pedestal. The peaks, recognized by the Root TSpectrum method [11], are marked
by red triangles. Right: number of collected photoelectrons versus the peak
ADC counts. The pedestal is excluded and a linear fit is superimposed
according to formula (3.1).

From the comparison of the ADC single-channel spectra of SiPM 3 and 4, it is
possible to check the performance of the adapter glued to the Winston cone.
For this check, the events were acquired when at least one channel was over
the threshold in both SiPMs. In such trigger configuration, the acquired
events are mainly due to cosmic particles entering both calorimeter modules,
resulting in a rate of $\approx$ 1.5 Hz. In Fig. 5 left, the two spectra are
displayed. It is evident that the light collection is higher for the coupling
with the adapter (SiPM 4). Indeed, using the function (3.1) and the parameters
of the fit in Fig. 4, right, the mean number of collected p.e. in these
distributions is $\sim 3.9$ for SiPM 3 and $\sim 5.4$ for SiPM 4. This is the
expected result considering the optical matching improvement due to the
adapter used for SiPM 4.

Figure 5: Left: the ADC single-channel spectrum for SiPM 3 (blue), compared
with that of SiPM 4 (red). The spectra are shown up to 2000 counts. Right:
average ADC counts versus arbitrary channel numbers. The average values are
calculated from the whole distribution.

This result is also confirmed by the observation that the average ADC counts
for all the channels of SiPM 4 are higher with respect to those of SiPM 3
(Fig. 5, right). The light collection is quite uniform and a range of
$\sim$100 is the spread of the average ADC counts for both SiPM 3 and SiPM 4.
Using the fit in Fig. 4, right, it is estimated that this spread corresponds
to $\pm$0.5 p.e. Therefore all channels have quite similar efficiency in the
same SiPM and the Winston cones homogenize the light signal. Using light
guides on both ends of each calorimeter module ensures that the SiPM
configuration is comparable to the PMT one in terms of light collection.
Moreover, it was verified that both SiPM 3 and SiPM 4 are well-suited for the
intended study, and they perform similarly.

##  4 Efficiency

The SiPM efficiency is measured by detecting cosmic muons at a stable lab
temperature of 23$\pm 1^{\circ}$C. A system of 4 scintillators, with a surface
of $\sim 2\times 7.5$ cm2 each, provides the external trigger. Two
scintillators are placed above (A and B green elements in Fig. 2) and two
below (C and D in Fig. 2) the calorimeter, overlapping with its central
modules. The trigger logic consists of a fired scintillator on the top (A or
B) and a fired scintillator on the bottom (C or D) in a time window of 30 ns.
The measured rate is 2.6 mHz as expected from the cosmic muon flux, taking
into account the narrow solid angle due to the trigger geometry. When a cosmic
muon activates this trigger, it is expected to pass through the stack of ECAL
modules, generating photons in the fibers that can be detected by SiPMs and
PMTs. Thus, during the SiPMs data acquisition, the PMT efficiency has also
been measured. Finally, we observe that the particle-crossing probability is
symmetrical with respect to the center of ECAL modules and the trigger does
not favor SiPM instead of PMT or vice versa.

The PMT signals are counted by a CAEN Quad Scaler module after they are shaped
with a threshold of 40 mV. The SiPM signals are managed by the CAEN FERS-
DT5202 board with two different outputs, both requiring the self-trigger and
validated by the external trigger. The used self-trigger is the OR-16 trigger
with a low threshold of 210 a.u., to collect also events with only 1 photon on
the whole SiPM array. When the validation from the external trigger is
satisfied, the counts are acquired with the CAEN Quad Scaler module and the
digital full information on the SiPM signals is collected by the data
acquisition program running on a desktop personal computer. Although the
requirements are identical, the counts from the outputs of the FERS board are
different. The events collected by the acquisition program (NDAQ) are
typically fewer than the shaped signals (kSiPM) sent directly to the Quad
Scaler as an effect of the DAQ board dead time. Indeed the board manages all
OR-16 triggers, so an event associated with the external trigger can be lost
if it arrives when the board is busy with the analog-to-digital conversion of
a previous event. On the contrary, this dead time does not affect the direct
output and the consequent counting by the scaler.

The sample collected by the DAQ is analyzed in order to identify and remove a
possible residual background due to the dark noise favored by the choice of a
very low threshold. Before the definition of a cut to clean the sample, it
must be taken into account that the SiPM array is studied as a unique
detector, and then the cut must be applied to the sum of the ADC of all
channels (ADC-sum). To investigate the SiPM noise, samples have been collected
requiring only the OR-16 trigger and not the validation of the external
trigger. The ADC-sum distribution of the sample collected in this condition
for SiPM 3 is shown in Fig. 6, left (red histogram). A clear peak is visible
at $\sim$2000 ADC with some events at higher values. In the same plot also the
distribution for events validated by the external trigger is reported. It is
evident the superposition of the two distributions at low values of the ADC-
sum. Similar results were obtained for the SiPM 4.

Figure 6: Sum of the ADC values (ADC-sum) for SiPM 3. Left: Distributions for
different trigger conditions in the range 0-15000 (more details in the text).
Right: Distribution with external trigger in the range 0-30000. The
superimposed fit is a combination of Landau and Gaussian functions (LanGaus).
The red part of the distribution is rejected requiring ADC-sum $>$ 3000.

On this basis, a reasonable cut to measure the SiPM efficiency is ADC-sum $>$
3000\. Another more stringent cut is suggested by the fit of the ADC-sum
distribution by means of a Landau function convoluted with a Gaussian (Fig. 6,
right) to take into account the multiplicity spread of collected photons and
the behavior of the detector. In this case, the samples can be reduced
requiring ADC-sum $>$ 5000\. Then the SiPM efficiency calculated from the
scaler counts ($\epsilon$) must be reduced accepting only the fraction of
events surviving the ADC-sum cut ($\epsilon^{\prime}$). The corrected SiPM
efficiency can be expressed as:

$E_{SiPM}=\epsilon\ \epsilon^{\prime}=\frac{k_{SiPM}}{N_{SiPM}}\
\frac{k_{DAQ}}{N_{DAQ}},$ (4.1)

where $k_{SiPM}$ is the number of SiPM events on the scaler, $N_{SiPM}$ is the
number of external triggers, $k_{DAQ}$ and $N_{DAQ}$ are respectively the
number of selected events and the total number of events in the sample
collected by the DAQ. On the other side, the PMT efficiency is calculated
simply from the scaler counts:

$E_{PMT}=\frac{k_{PMT}}{N_{PMT}},$ (4.2)

where the meaning of $k_{PMT}$ and $N_{PMT}$ is obvious. The results of the
efficiency measurement are shown in Table 1 where the Clopper-Pearson method
[13] has been used to estimate the errors and 3000 is the ADC-sum threshold to
accept SiPM events. It can be observed that the higher average signal for SiPM
4 (see again Fig. 5) is related to a higher value of $\epsilon$ (96.7%) with
respect to SiPM 3 (95.0%). This difference is balanced by a higher
contribution of noise events and a lower value of $\epsilon^{\prime}$ (93.9%)
compared to SiPM 3 (95.5%). The resulting efficiencies
($\epsilon\epsilon^{\prime}$) for SiPM 3 and 4 are finally very close as a
demonstration of the method reliability. As an additional test, a different
self-trigger condition was set in the CAEN FERS-DT5202 board by increasing the
threshold to 230. The result was a reduction of $\epsilon$ and an enhancement
of $\epsilon^{\prime}$ without any significant change in the final result.

Using the stricter threshold suggested by the LanGaus fit (ADC-sum $>$ 5000),
the efficiencies are 88.34% and 89.16% for SiPM 3 and 4, respectively, with
errors of the same magnitude as in Table 1. In conclusion, the efficiencies of
PMTs and SiPMs are very close, with the PMT efficiencies being slightly higher
than the SiPM ones.

Table 1: Efficiency measurements for PMTs and SiPMs. The kDAQ value is estimated requiring ADC-sum $>$ 3000. PMT | kPMT | NPMT |  |  | EPMT (%)
---|---|---|---|---|---
3 | 7010 | 7615 |  |  | $92.06^{+0.14}_{-0.15}$
4 | 6739 | 7392 |  |  | $91.17^{+0.15}_{-0.16}$
SiPM | kSiPM | NSiPM | kDAQ | NDAQ | ESiPM (%)
3 | 4324 | 4552 | 2999 | 3141 | $90.70^{+0.22}_{-0.23}$
4 | 4820 | 4982 | 2680 | 2855 | $90.82^{+0.23}_{-0.25}$

##  5 Timing resolution

The timing resolution of the photodetectors has been measured using the rising
time at 50% of the signal amplitude. The experimental setup is shown in Fig.
7, where the ECAL modules 3 and 4 are coupled with PMTs on one side and SiPMs
on the opposite side, both connected to the oscilloscope.

Figure 7: Setup used to measure the timing resolution of SiPMs and PMTs. Two
ECAL modules, coupled with PMTs on one side and SiPMs on the opposite side,
are used for this measurement. The signal is acquired by the oscilloscope,
which also provides the timing information.

In the case of PMTs, the signals from modules 3 and 4 are compared in order to
measure $\Delta t$. For SiPMs, the measurement is carried out considering
either two channels from the same array (SiPM 3 $\alpha$ and SiPM 3 $\beta$ or
SiPM 4 $\alpha$ and SiPM 4 $\beta$, with $\alpha$ and $\beta$ indicating a
specific channel) or one channel per array on modules 3 and 4. The signals
generated in coincidence by cosmic rays have been acquired using the
oscilloscope without amplification and with a 40 GHz sampling rate, employing
the constant fraction triggering technique as the time pick-off method. A
typical signal amplitude is of several hundreds of mV for PMT and several tens
of mV for SiPM, as can be seen in Fig. 8, where the signals (as they appear on
the oscilloscope display) are shown.

Figure 8: Coincident signals on the oscilloscope display read using two SiPMs
(left) or two PMTs (right).

The PMT signal covers a shorter time interval ($\sim$20 ns) with respect to
the SiPM one ($\sim$80 ns). Furthermore, it can be observed that the signal
shape can be approximated with a triangle with the base indicating the
duration. The area of the signals with respect to their amplitudes is shown in
Fig. 9 for SiPMs and PMTs.

Figure 9: Area versus amplitude of the signal for SiPM 4 (left) and PMT 4
(right). The events with red markers are rejected (more detail in the text).
Similar plots are available for SiPM 3 and PMT 3.

Both plots show a linearity of area versus amplitude meaning a quasi-constant
duration, as expected. Thus the events of interest have been selected by
fitting the linear region on these plots and by defining a selection range
around the fit line. The accepted events are shown with blue markers, while
the red ones are rejected. The latter are events with saturated signals on at
least one channel. For SiPMs, also an electromagnetic noise 333The source of
this noise was uniquely identified but it was not eliminable. was collected
because of a lack of shielding of the custom electronic board used to pick up
the signals. Such noisy signal was characterized by an area close to zero, has
been removed off-line and did not affect the PMT signals.

The PMTs exhibit the presence of two families of events, identified by bands
with different slopes (Fig. 9, right). Considering the signal triangular
shape, it results $area\sim(a/2)\times amplitude$, where the quantity $a$ is a
rough estimate of the duration. It follows that the less slanted family of
events is characterized by a faster development. The separation is more
evident in Fig. 10, where the $a$-quantity from both PMTs are shown together.
The short signal duration is attributed to the Cherenkov effect occurring in
the lucite light guides near the PMTs. The longer signal has a higher number
of events, and matches the characteristics of the scintillation light produced
by particles crossing the calorimeter.

Figure 10: Estimate of the signal duration for PMT 3 and PMT 4. The two
separated clusters of events are attributed to photons generated by the
Cherenkov (magenta) or scintillation (blue) effect.

The time difference at 50% of the signals is shown in Fig. 11 for two channels
of the same SiPM matrix, and in Fig. 12 for SiPMs (left) and PMTs (right)
placed on different ECAL modules (3 and 4), after the event selection
described above. For PMTs only the scintillation light is taken into account.

Figure 11: Time difference for two channels of SiPM 3 (left) and SiPM 4
(right). The Gaussian fit (in red) is superimposed.

Figure 12: Left: time difference for SiPM 3 and SiPM 4. Right: time difference
for PMT 3 and PMT 4. The distributions are fitted with a Gaussian function (in
red).

For two channels of the same SiPM array (Fig. 11), the distribution mean value
is due to an offset ($\sim 128$ ps), introduced by the custom electronic board
used to pick up the signals. Since each SiPM in the array gives an independent
signal readout, the intrinsic timing resolution ($\sigma_{t}$) of the single
SiPM channel is $\sigma_{t}=\sigma_{\Delta t}/\sqrt{2}$.

In the case of detectors coupled with different ECAL modules (Fig. 12), the
mean value depends on the average time of flight of a relativistic particle
from module 3 to module 4 and, for the SiPMs, also on the electronics offset
and the delay introduced by the adapter on SiPM 4. Consequently, the
$\sigma_{\Delta t}$ of these distributions is affected by the additional
jitter due to the various particle paths (geometrical jitter). The intrinsic
timing resolution is estimated by simulating the experimental setup and a muon
flux proportional to cos${}^{2}\theta$, with $\theta$ being the zenith angle
[14]. The estimation consists of looking for the intrinsic timing resolution
which reproduces the measured $\sigma_{\Delta t}$ when combined with the
geometrical jitter. All the measurements of the timing resolution are
summarized in Table 2. It can be concluded that the PMTs exhibit a slightly
better timing resolution than the SiPM.

Table 2: Timing resolution for SiPMs and PMTs. The $\sigma_{\Delta t}$ and its error are derived from the fit of distributions in Figs. 11 and 12. The $\sigma_{t}$ is the intrinsic timing resolution of the single detector. |  $\sigma_{\Delta t}$ (ps) |  $\sigma_{t}$ (ps)
---|---|---
SiPM 3 $\alpha$ \- $\beta$ | 372$\pm$8 | 263$\pm$6
SiPM 4 $\alpha$ \- $\beta$ | 341$\pm$8 | 241$\pm$6
SiPM 3$\alpha$ \- SiPM 4$\alpha$ | 419$\pm$7 | 269$\pm$5
PMT 3 - PMT 4 | 350$\pm$5 | 217$\pm$4

The same measurement procedure was repeated at four different temperatures in
the range from $\sim 17^{\circ}$C to $\sim 27$°C, measured by a sensor placed
near the SiPM matrix. It has been verified that the SiPM timing resolution is
not significantly affected by the temperature variation in this range.

The $\Delta t$ distribution for Cherenkov events detected in the PMTs is shown
in Fig. 13, fitted with a Gaussian function. The mean value and the
$\sigma_{\Delta t}$ are smaller than what is obtained when the scintillation
light is considered due to intrinsic characteristics of the scintillation
process. Specifically, in this case, $\sigma_{\Delta t}$ = 109 $\pm$ 3 ps,
which results in a timing resolution $\sigma_{t}\sim$ 70 ps. The Cherenkov
effect was not observed in the SiPMs collected data. This is probably related
to the fact that the SiPM signal is not amplified.

Figure 13: Time difference for PMT 3 and PMT 4, considering only Cherenkov
light.

The timing resolution of the KLOE calorimeter with PMT read-out (used in this
work) depends on the deposited energy ($E$), according to the following
formula [3]:

$\sigma_{t}\sim 54~{}ps/\sqrt{E~{}(\rm{GeV})}.$ (5.1)

A similar behavior has been observed in this study for SiPMs and PMTs taking
into account the signal amplitude instead of the deposited energy. The results
are shown in Fig. 14, and are well described by the formula:

$\sigma_{\Delta t}=\tau/\sqrt{signal~{}(\rm{mV})},$ (5.2)

where $\tau$ is the fit parameter. Although the value of $\tau$ cannot be
compared with 54 ps because signal (in mV) and energy (in GeV) are different
quantities, they are typically linearly related and the similarity of
functions (5.1) and (5.2) confirms the dependence of timing resolution on
energy (and signal). The different values of $\tau$ for SiPM and PMT is not a
surprise because the signals are completely different for the two devices, as
the values of $\sigma_{\Delta t}$.

Figure 14: Timing resolution as a function of the signal for SiPMs (left) and
PMTs (right). The red curve represents the fit function reported in the text
(see formula 5.2).

In conclusion, the timing resolution was computed and compared for PMTs and
SiPMs. When the scintillation light is considered, the two photodetectors show
similar results, with the PMT performing slightly better. Additionally, PMTs
are also sensitive to Cherenkov light exhibiting a smaller timing resolution
than the scintillation case. This indicates that the intrinsic timing
resolution of PMTs can be even better, while no further improvement is
appreciable for the SiPMs.

###  5.1 Consistency check on the measurement method

Some checks were made during the process, in order to validate the performed
measurements. The first consistency check is shown in Fig. 15, where the
amplitude of signals taken from two separate channels of the same SiPM matrix
are compared. As expected, the distribution of events is well described by a
bisector. This result confirms the uniformity of the photon collection event-
by-event on the SiPM array. This uniformity was already observed on
statistical basis (see Fig. 5, right).

Figure 15: Signal amplitude of channel $\alpha$ versus amplitude of channel
$\beta$ of the same SiPM array.

Another check can be performed by looking at the difference in time
($t_{4}-t_{3}$) and signal amplitude ($signal_{4}-signal_{3}$) on different
ECAL modules. These quantities are related because both increase with the
track slope. The complete description of this dependence must take into
account the exponential reduction of the number of detected photons (and the
collected signal) with the path in the calorimeter bar, depending on the
attenuation length ($\sim 325$ cm for Pol.Hi.Tech-0046 fibers [15]). Since the
light path is on average much smaller than the attenuation length, it is
possible to approximate this exponential dependence with a linear one. This is
confirmed from the profiles in Fig. 16 where a linear relation between
$\langle\Delta signal\rangle$ and $\langle\Delta t\rangle$ is visible for
SiPMs and PMTs.

Figure 16: Time difference vs signal amplitude difference (profile). Left:
SiPM. Right: PMT.

##  6 Conclusions

A piece of the KLOE calorimeter has been equipped with SiPMs and PMTs in order
to compare their performance for a potential application in an element (SAND)
of the Near Detector complex of the future DUNE experiment. The PMT coupling
was optimized in the KLOE calorimeter and it was kept unchanged for this
study. Various optical couplings have been tested for adapting the SiPMs with
the KLOE light guides and it was verified that the partial coverage of the
light guide does not prevent reaching a proper readout of the signal.
Efficiency and timing resolution have been measured and compared for both
detectors. It was found that the SiPM noise cannot be neglected in the
efficiency estimate. Specifically, the noise rejection implies a non-
negligible reduction of the possible SiPM efficiency making it lower by a few
percent with respect to the PMT ones. For the timing resolution, it was
observed that PMTs reach a value lower than 220 ps for the scintillation
signal, while SiPMs exhibit a resolution above 240 ps. Even if the difference
is small, PMTs seem to perform better.

In conclusion, the difficulties in coupling SiPMs to ECAL without deep
mechanical changes, the lack of improvement, the cost, and the necessary
commissioning time, advise against the substitution of the 4880 available and
tested PMTs with new SiPMs in the KLOE calorimeter to be reused in the Near
Detector complex in DUNE. Nonetheless, the results from this study do not
exclude the possible use of SiPMs for other calorimetry applications.

## Acknowledgments

We would like to thank Stefano Miscetti (INFN, Laboratori Nazionali di
Frascati) who delivered to us the stack of lead foils and scintillating
fibers. We thank also Antonio Di Domenico (Sapienza, Rome) and Alessandro
Montanari (INFN, Sezione di Bologna) for their fruitful comments and
suggestions.

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