Narrow Results

We study the endomorphisms and derivations of an infinite-dimensional cyclic Leibniz algebra. Among others it was found that if $L$ is a cyclic infinite-dimensional Leibniz algebra over a field $F$, then the group of all automorphisms of $L$ is isomorphic to a multiplicative group of the field $F$. The description of an algebra of derivations of a cyclic infinite-dimensional Leibniz algebra has been obtained.

Given a complete rewriting system R on X and a subset X₀ of X⁺ satisfying certain conditions, we present a complete rewriting system for the submonoid of M(X;R) generated by X₀. The obtained result will be applied to the group of units of a monoid satisfying H₁ = D₁. On the other hand we prove that all maximal subgroups of a monoid defined by a special rewriting system are isomorphic.

A monoid M is called surjunctive if every injective cellular automata with finite alphabet over M is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all residually finite monoids, all finitely generated linear monoids, and all cancellative one-sided amenable monoids are surjunctive. We also prove that every limit of marked surjunctive monoids is itself surjunctive. On the other hand, we show that the bicyclic monoid and, more generally, all monoids containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive.

We prove that the singular braid monoid of [2] and [5] embeds in a group. This group has a geometric interpretation as singular braids with two type of singularities which cancel.

It is known, since the early 2000s, that the Catalan monoid $C_n$ generated by $n$ elements is finitely based if and only if $n\le 3$. The main goal of this paper is to prove that the involution monoid $(C_3,{}^{*})$ is non-finitely based. Therefore, in contrast, combining with previous results yields that the involution Catalan monoid $(C_n,{}^{*})$ is finitely based if and only if $n=1$.

The Kiselman monoid $K_n$ generated by $n$ elements is also considered. Although the semigroups $C_n$ and $K_n$ have recently been shown to satisfy the same identities, it is unknown if the same result holds when they are considered as involution semigroups. Nevertheless, it is deduced from the main result that the involution Kiselman monoid $(K_n,{}^{*})$ is finitely based if and only if $n=1$.

Hecke–Kiselman monoids ${\mathrm{\text{HK}}}_{\mathrm{\Theta }}$ and their algebras $K[{\mathrm{\text{HK}}}_{\mathrm{\Theta }}]$, over a field $K$, associated to finite oriented graphs $\mathrm{\Theta }$ are studied. In the case $\mathrm{\Theta }$ is a cycle of length $n\ge 3$, a hierarchy of certain unexpected structures of matrix type is discovered within the monoid $C_n={\mathrm{\text{HK}}}_{\mathrm{\Theta }}$ and this hierarchy is used to describe the structure and the properties of the algebra $K[C_n]$. In particular, it is shown that $K[C_n]$ is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand–Kirillov dimension one. This is used to characterize all Noetherian algebras $K[{\mathrm{\text{HK}}}_{\mathrm{\Theta }}]$ in terms of the graphs $\mathrm{\Theta }$. The strategy of our approach is based on the crucial role played by submonoids of the form $C_n$ in combinatorics and structure of arbitrary Hecke–Kiselman monoids ${\mathrm{\text{HK}}}_{\mathrm{\Theta }}$.

A variety of algebras is called limit if it is nonfinitely-based but all its proper subvarieties are finitely-based. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with commuting idempotents.

Factorizations of monoids are studied. Two necessary and sufficient conditions in terms of the so-called descent 1-cocycles for a monoid to be factorized through two submonoids are found. A full classification of those factorizations of a monoid whose one factor is a subgroup of the monoid is obtained. The relationship between monoid factorizations and non-abelian cohomology of monoids is analyzed. Some applications of semi-direct product of monoids are given.

In this paper we prove that the pseudovariety of Abelian groups is hyperdecidable and moreover that it is completely tame. This is a consequence of the fact that a system of group equations on a free Abelian group with certain rational constraints is solvable if and only if it is solvable in every finite quotient.

We continue our program of extending key techniques from geometric group theory to semigroup theory, by studying monoids acting by isometric embeddings on spaces equipped with asymmetric, partially defined distance functions. The canonical example of such an action is a cancellative monoid acting by translation on its Cayley graph. Our main result is an extension of the Švarc–Milnor lemma to this setting.

Leech’s (co)homology groups of finite cyclic monoids are computed.

We give necessary and sufficient conditions for the efficiency of the direct product of finitely many finite monogenic monoids.

In this paper, the endomorphism monoid of circulant complete graph K(n,3) is explored explicitly. It is shown that AutK(n,3)) =D_n, the dihedral group of degree n. It is also shown that K(n,3) is unretractive when 3 does not divide n, End(K(3m,3)) =qEnd(K(3m,3)), sEnd(K(3m,3)) =Aut(K(3m,3)) and K(3m,3) is endomorphism-regular. The structure of End(K(3m,3)) is characterized and some enumerative problems concerning End(K(n,3)) are solved.

Let L_S denote the language of (right) S-acts over a monoid S and let Σ_S be a set of sentences in L_S which axiomatises S-acts. A general result of model theory says that Σ_S has a model companion, denoted by T_S, precisely when the class of existentially closed S-acts is axiomatisable and in this case, T_S axiomatises . It is known that T_S exists if and only if S is right coherent. Moreover, by a result of Ivanov, T_S has the model-theoretic property of being stable.

In the study of stable first order theories, superstable and totally transcendental theories are of particular interest. These concepts depend upon the notion of type: we describe types over T_S algebraically, thus reducing our examination of T_S to consideration of the lattice of right congruences of S. We indicate how to use our result to confirm that T_S is stable and to prove another result of Ivanov, namely that T_S is superstable if and only if S satisfies the maximal condition for right ideals. The situation for total transcendence is more complicated but again we can use our description of types to ascertain for which right coherent monoids S we have that T_S is totally transcendental and is such that the U-rank of any type coincides with its Morley rank.