We study congruences on the partial automorphism monoid of a finite rank free group action. We give a decomposition of a congruence on this monoid into a Rees congruence, a congruence on a Brandt semigroup and an idempotent separating congruence. The constituent parts are further described in terms of subgroups of direct and semidirect products of groups. We utilize this description to demonstrate how the number of congruences on the partial automorphism monoid depends on the group and on the rank of the action.
The twisted partition monoid $\mathcal{P}_n^Φ$ is an infinite monoid obtained from the classical finite partition monoid $\mathcal{P}_n$ by taking into account the number of floating components when multiplying partitions. The main result of this paper is a complete description of the congruences on $\mathcal{P}_n^Φ$. The succinct encoding of a congruence, which we call a C-pair, consists of a sequence of $n+1$ congruences on the additive monoid $\mathbb{N}$ of natural numbers and a certain $(n+1)\times\mathbb{N}$ matrix. We also give a description of the inclusion ordering of congruences in t…
A function on an algebra is congruence preserving if, for any congruence, it maps congruent elements to congruent elements. We show that, on a free monoid generated by at least 3 letters, a function from the free monoid into itself is congruence preserving %nonmonogenic if and only if it is of the form $x \mapsto w_0 x w_1 \cdots w_{n-1} x w_n$ for some finite sequence of words $w_0,\ldots,w_n$. We generalize this result to functions of arbitrary arity. This shows that a free monoid with at least three generators is a (noncommutative) affine complete algebra. Up to our knowledge, it is the fir…
Given a poset $P$ with at least two elements and a group $G$, there exists a selfdual lattice of length 16 such that the collection of its principal congruences is order isomorphic to $P$ while its automorphism group to $G$.
We build on the recent characterisation of congruences on the infinite twisted partition monoids $\mathcal{P}_{n}^Φ$ and their finite $d$-twisted homomorphic images $\mathcal{P}_{n,d}^Φ$, and investigate their algebraic and order-theoretic properties. We prove that each congruence of $\mathcal{P}_{n}^Φ$ is (finitely) generated by at most $\lceil\frac{5n}2\rceil$ pairs, and we characterise the principal ones. We also prove that the congruence lattice $\textsf{Cong}(\mathcal{P}_{n}^Φ)$ is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending cha…
We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I->M onto the minimal ideal, a congruence on M, and a normal subgroup of a maximal subgroup outside I.
Automorphism groups of direct products of multiplicative monoids of certain rings
University of Stellenbosch（斯泰伦博斯大学）；NITheCS (National Institute for Theoretical and Computational Sciences)（国家理论与计算科学研究所）；University of Minho（米尼奥大学）；CMAT (Centre of Mathematics)（数学中心）
The partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup $S$ with no isolated nontrivial subgroups is lattice determined "modulo semilattices" and if $T$ is an inverse semigroup whose partial automorphism monoid is isomorphic to that of $S$, then either $S$ and $T$ are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if $T$ is any semigroup and the inverse monoids consisting of all isom…
We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $PB_X$, where $X$ is an arbitrary infinite set, and also of the lattices formed by all such congruences. Our results complement those from a recent article of East, Mitchell, Ruskuc and Torpey, which deals with the finite case. As a consequence of our classification result, we show that the congruence lattices of $P_X$ and $PB_X$ are isomorphic to each other, and are distributive and well quasi-ordered. We also calculate the smallest number of pairs of partitions required to generate a…
One of the results in our article, which appeared in Publ. Mat. 65 (2021), 499--528, is that the structure monoid $M(X,r)$ of a left non-degenerate solution $(X,r)$ of the Yang-Baxter Equation is a left semi-truss, in the sense of Brzeziński, with an additive structure monoid that is close to being a normal semigroup. Let $η$ denote the least left cancellative congruence on the additive monoid $M(X,r)$. It is then shown that $η$ also is a congruence on the multiplicative monoid $M(X,r)$ and that the left cancellative epimorphic image $\bar{M}=M(X,r)/η$ inherits a semi-truss structure and thus …
Dilworth's theorem. Every finite distributive lattice $D$ can be represented as the congruence lattice of a finite lattice $L$. We want: Every finite distributive lattice $D$ can be represented as the congruence lattice of a nice finite lattice $L$. nice = sectionally complemented, uniform, semimodular, given automorphism group, regular, uniform, isoform
The algebraic monoid structure of an incidence algebra is investigated. We show that the multiplicative structure alone determines the algebra automorphisms of the incidence algebra. We present a formula that expresses the complexity of the incidence monoid with respect to the two sided action of its maximal torus in terms of the zeta polynomial of the poset. In addition, we characterize the finite (connected) posets whose incidence monoids have complexity $\leq 1$. Finally, we determine the covering relations of the adherence order on the incidence monoid of a star poset.
Motivated by a recent paper of G. Grätzer, a finite distributive lattice $D$ is said to be fully principal congruence representable if for every subset $Q$ of $D$ containing $0$, $1$, and the set $J(D)$ of nonzero join-irreducible elements of $D$, there exists a finite lattice $L$ and an isomorphism from the congruence lattice of $L$ onto $D$ such that $Q$ corresponds to the set of principal congruences of $L$ under this isomorphism. Based on earlier results of G. Grätzer, H. Lakser, and the present author, we prove that a finite distributive lattice $D$ is fully principal congruence represent…
A new result of G. Czédli states that for an ordered set $P$ with at least two elements and a group $G$, there exists a bounded lattice $L$ such that the ordered set of principal congruences of $L$ is isomorphic to $P$ and the automorphism group of $L$ is isomorphic to $G$. I provide an alternative proof utilizing a result of mine with J. Sichler from the late 1960-s.
A partial automorphism of a semigroup $S$ is any isomorphism between its subsemigroups, and the set all partial automorphisms of $S$ with respect to composition is the inverse monoid called the partial automorphism monoid of $S$. Two semigroups are said to be $\cPA$-isomorphic if their partial automorphism monoids are isomorphic. A class $\K$ of semigroups is called $\cPA$-closed if it contains every semigroup $\cPA$-isomorphic to some semigroup from $\K$. Although the class of all inverse semigroups is not $\cPA$-closed, we prove that the class of inverse semigroups, in which no maximal isola…
In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids $IEnd(P_n)$ and $PAut(P_n)$ of all injective partial endomorphisms and of all partial automorphisms of the undirected path $P_n$ with $n$ vertices. We also describe Green's relations of $PAut(P_n)$ and $IEnd(P_n)$ and calculate their cardinals.
The CREAM GAP package computes automorphisms, congruences, endomorphisms and subalgebras of algebras with an arbitrary number of binary and unary operations; it also decides if between two such algebras there exists a monomorphism, an epimorphism, an isomorphism or if one is a divisor of the other. Thus it finds those objects for almost all algebras used in practice (groups, quasigroups in their various signatures, semigroups possibly with many unary operations, fields, semi-rings, quandles, logic algebras, etc). As a one-size-fits-all package, it only relies on universal algebra theorems, wit…
This paper presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley--Lieb and Jones partitions; linear and projec…
In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automor…
