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For a group G acting on a set X, let End G (X) be the monoid of all G-equivariant transformations, or G-endomorphisms, of X, and let Aut G (X) be its group of units. After discussing few basic results in a general setting, we focus on the case when G and X are both finite in order to determine the smallest cardinality of a set W ⊆ End G (X) such that W ∪ Aut G (X) generates End G (X) ; this is known in semigroup theory as the relative rank of End G (X) modulo Aut G (X) . [ABSTRACT FROM AUTHOR]

We study some varieties of flat nil-semirings and provide a sufficient condition for a finite flat semiring to be nonfinitely based. As applications, we show that a finite flat nil-semiring of the form S c (W) is finitely based if and only if W consists of cubes of letters and words of length at most 2. Also, we prove that the finite flat nil-semirings S(W) are nonfinitely based for some sets W of special words in X + . This affirmatively answers two problems raised by Jackson et al. (J Algebra 611:211–245, 2022). [ABSTRACT FROM AUTHOR]

We characterize the monoid of endomorphisms of the semigroup of all oriented full transformations of a finite chain, as well as the monoid of endomorphisms of the semigroup of all oriented partial transformations and the monoid of endomorphisms of the semigroup of all oriented partial permutations of a finite chain. Characterizations of the monoids of endomorphisms of the subsemigroups of all orientation-preserving transformations of the three semigroups aforementioned are also given. In addition, we compute the number of endomorphisms of each of these six semigroups. [ABSTRACT FROM AUTHOR]

We exhibit a faithful representation of the stylic monoid of every finite rank as a monoid of upper unitriangular matrices over the tropical semiring. Thus, we show that the stylic monoid of finite rank n generates the pseudovariety J n , which corresponds to the class of all piecewise testable languages of height n, in the framework of Eilenberg's correspondence. From this, we obtain the equational theory of the stylic monoids of finite rank, show that they are finitely based if and only if n ≤ 3 , and that their identity checking problem is decidable in linearithmic time. We also establish connections between the stylic monoids and other plactic-like monoids, and solve the finite basis problem for the stylic monoid with involution. [ABSTRACT FROM AUTHOR]

Young tableaux are combinatorial objects whose construction can be achieved from words over a finite alphabet by row or column insertion as shown by Schensted sixty years ago. Recently Abram and Reutenauer studied the action of the free monoid on the set of columns defined by the famous insertion algorithm. Since the number of columns is finite, this action yields a finite transformation monoid. Here we consider the action on the set of rows. We investigate the corresponding infinite monoid in the case of a 3 letter alphabet. In particular we show that it is the quotient of the free monoid relative to a congruence generated by the classical Knuth rules plus a unique extra rule. [ABSTRACT FROM AUTHOR]

A C-monoid is a suitably defined submonoid of a factorial monoid with finite reduced class semigroup. This monoid plays a key role in an arithmetical investigation of a large class of Mori domains. It is well understood that a C-monoid is Krull if and only if the reduced class semigroup coincides with the (v-)class group of a Krull monoid, and the arithmetic of a Krull monoid can be determined by the structure of its (v-)class group. The finiteness of the reduced class semigroup allows us to prove the similar arithmetical finiteness for a general C-monoid to results known in the Krull case. Recently, the algebraic structure of the reduced class semigroup has begun to be studied for a non-Krull C-monoid. Every Krull monoid is a root-closed weakly Krull Mori monoid, and under the mild conditions, a root-closed weakly Krull Mori monoid is a C-monoid. In this paper, we study the structure of a root-closed weakly Krull Mori monoid and of its class semigroup. [ABSTRACT FROM AUTHOR]

Let Q be an inverse semigroup. A subsemigroup S of Q is a left I-order in Q and Q is a semigroup of left I-quotients of S if every element in Q can be written as a - 1 b , where a , b ∈ S and a - 1 is the inverse of a in the sense of inverse semigroup theory. If we insist on being able to take a and b to be R -related in Q we say that S is straight in Q and Q is a semigroup of straight left I-quotients of S. We give a set of necessary and sufficient conditions for a semigroup to be a straight left I-order. The conditions are in terms of two binary relations, corresponding to the potential restrictions of R and L from an oversemigroup, and an associated partial order. Our approach relies on the meet structure of the L -classes of inverse semigroups. We prove that every finite left I-order is straight and give an example of a left I-order which is not straight. [ABSTRACT FROM AUTHOR]

We observe that for each n ≥ 2 , the identities of the stylic monoid with n generators coincide with the identities of n-generated monoids from other distinguished series of J -trivial monoids studied in the literature, e.g., Catalan monoids and Kiselman monoids. This solves the Finite Basis Problem for stylic monoids. [ABSTRACT FROM AUTHOR]

The free monoid A ∗ on a finite totally ordered alphabet A acts at the left on columns, by Schensted left insertion. This defines a finite monoid, denoted Styl (A) and called the stylic monoid. It is canonically a quotient of the plactic monoid. Main results are: the cardinality of Styl (A) is equal to the number of partitions of a set on | A | + 1 elements. We give a bijection with so-called N-tableaux, similar to Schensted's algorithm, explaining this fact. Presentation of Styl (A) : it is generated by A subject to the plactic (Knuth) relations and the idempotent relations a 2 = a , a ∈ A . The canonical involutive anti-automorphism on A ∗ , which reverses the order on A, induces an involution of Styl (A) , which similarly to the corresponding involution of the plactic monoid, may be computed by an evacuation-like operation (Schützenberger involution on tableaux) on so-called standard immaculate tableaux (which are in bijection with partitions). The monoid Styl (A) is J-trivial, and the J-order of Styl (A) is graded: the co-rank is given by the number of elements in the N-tableau. The monoid Styl (A) is the syntactic monoid for the the function which associates to each word w ∈ A ∗ the length of its longest strictly decreasing subword. [ABSTRACT FROM AUTHOR]

Dibands and tribands are generalizations of bands. Zhuchok asked two open problems on constructing a free diband in 2017 and on constructing a free triband in 2019. We solve these two open problems and calculate the cardinality of the free diband (resp. free triband) generated by an arbitrary finite set. Moreover, we prove that the word problem for free dibands (resp. free tribands) is solvable. [ABSTRACT FROM AUTHOR]

We characterise the respective semigroups of mappings that preserve, or that preserve or reverse orientation of a finite cycle, in terms of their actions on oriented triples and oriented quadruples. This leads to a proof that the latter semigroup coincides with the semigroup of all mappings that preserve intersections of chords on the corresponding circle. [ABSTRACT FROM AUTHOR]

Let X be a nonempty set and T X the full transformation semigroup on X. For a partition P of X, we study semigroups T (X , P) = { f ∈ T X ∣ (∀ X i ∈ P) (∃ X j ∈ P) X i f ⊆ X j } , Σ (X , P) = { f ∈ T (X , P) ∣ (∀ X i ∈ P) X f ∩ X i ≠ ∅ } , and Γ (X , P) = { f ∈ T X ∣ (∀ X i ∈ P) (∃ X j ∈ P) X i f = X j } under composition. We give necessary and sufficient conditions for Γ (X , P) to be the semigroup of all closed selfmaps on X endowed with the topology having P as a basis. For finite X, we characterize unit-regular elements of T (X , P) and Σ (X , P) . We discuss set inclusions between Γ (X , P) and certain semigroups of selfmaps that preserve P . We characterize idempotents and regular elements of Γ (X , P) . For finite X, we also prove that all regular elements of Γ (X , P) are unit-regular. We finally count the number of elements, idempotents, and regular elements of Γ (X , P) for finite X. [ABSTRACT FROM AUTHOR]

Let I n be the symmetric inverse semigroup on X n = { 1 , 2 , ⋯ , n } , and let O D P n be the subsemigroup of I n consisting of all order-preserving partial isometries of X n . Let Y = { 1 , 2 , ⋯ , r } be a non-empty subset of X n . Define O D P I n , r = { α ∈ O D P n : im α ⊆ Y }. In this paper, we give a necessary and sufficient condition for O D P I n , r to be an inverse semigroup and characterize its Green relations. Moreover, the cardinality and the rank of O D P I n , r are investigated. [ABSTRACT FROM AUTHOR]

We construct a finite convergent semi-quadratic presentation for the Chinese monoid by adding column generators and using combinatorial properties of insertion algorithms on Chinese staircases. We extend this presentation into a coherent one whose generators are columns, rewriting rules are defined by insertion algorithms, and whose syzygies are defined as relations among insertion algorithms. Such a coherent presentation is used for representations of Chinese monoids, in particular, it is a way to describe actions of Chinese monoids on categories. [ABSTRACT FROM AUTHOR]

We investigate Johnson pseudo-contractibility and pseudo-contractibility of Clifford semigroup algebras. We show that, for a Clifford semigroup S, if ℓ 1 (S) has a central approximate identity in c 00 (S) , then ℓ 1 (S) is (Johnson) pseudo-contractible if and only if E(S) is locally finite and each maximal subgroup of S is (amenable) finite, respectively. As an application, we characterize Johnson pseudo-contractibility and pseudo-contractibility of ℓ 1 (S) , where S is a commutative semigroup, a band semigroup, or an inverse semigroup with totally ordered idempotents set. [ABSTRACT FROM AUTHOR]

We classify all partial equivalence relations according to their endotype with respect to endotopisms and compute the cardinalities of endotopism semigroups (strong endotopism monoids, autotopism groups) of partial equivalence relations on a finite set. [ABSTRACT FROM AUTHOR]

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right étale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A subsemiring S of R is called a positive semiring provided that S consists of nonnegative numbers and 1 ∈ S . Here we study factorizations in both the additive monoid (S , +) and the multiplicative monoid (S \ { 0 } , ·) . In particular, we investigate when, for a positive semiring S, both (S , +) and (S \ { 0 } , ·) have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP ⇒ BFP and FFP ⇒ BFP ⇒ ACCP ⇒ atomicity holds. Here we construct classes of positive semirings wherein both the additive and multiplicative structures satisfy each of these properties, and we also give examples to show that, in general, none of the implications in the previous chain is reversible. [ABSTRACT FROM AUTHOR]

The variety generated by the Brandt semigroup B 2 can be defined within the variety generated by the semigroup A 2 by the single identity x 2 y 2 ≈ y 2 x 2 . Edmond Lee asked whether or not the same is true for the monoids B 2 1 and A 2 1 . We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of A 2 1 that satisfy x 2 y 2 ≈ y 2 x 2 and contain B 2 1 . A further consequence is that the variety of B 2 1 cannot be defined within the variety of A 2 1 by any finite system of identities. Continuing downward, we then turn to subvarieties of B 2 1 . We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity x 2 y ≈ y x 2 and containing the monoid M (z ∞) , where z ∞ denotes the infinite limit of the Zimin words z 0 = x 0 , z n + 1 = z n x n + 1 z n . [ABSTRACT FROM AUTHOR]

We show that for any weakly cancellative uniformly locally finite inverse semigroup S such that l 1 (S) is ideally Connes amenable, for each D -classes D of S, E(D) is finite, where E(D) is the set of idempotents of D. This is similar to a theorem of Duncan and Paterson, that says if l 1 (S) is amenable then E(D) is finite. Also we study ideal Connes amenability of l 1 -Munn algebras. [ABSTRACT FROM AUTHOR]

It has been known for a long time that every finite orthodox completely regular semigroup has a finite basis of identities, and that every finite central completely simple semigroup has a finite basis of identities. In the present paper, a common generalization of these two facts is established. It is shown that every finite central locally orthodox completely regular semigroup has a finite basis of identities. The proof of this latter fact which is presented in this paper employs significantly the celebrated theorem of Libor Polák on the structure of the lattice of all varieties of completely regular semigroups. [ABSTRACT FROM AUTHOR]

Two semigroups are called Morita equivalent if the categories of firm right acts over them are equivalent. We prove that every semigroup is Morita equivalent to its subsemigroup consisting of all products of n factors. Using this we show that a finite semigroup is Morita equivalent to its largest factorisable subsemigroup. It follows that two finite semigroups are Morita equivalent if and only if their Cauchy completions are equivalent categories. Since these categories are finite, the problem of Morita equivalence of finite semigroups is decidable. [ABSTRACT FROM AUTHOR]

A semigroup S is called a Δ -semigroup if the lattice of its congruences forms a chain relative to the inclusion. A local automorphism of a semigroup S is defined as an isomorphism between its two subsemigroups. The set of all local automorphisms of a semigroup S relative to the operation of composition forms an inverse monoid of local automorphisms. We present a classification of all finite semigroups for which the inverse monoid of local automorphisms is a Δ -semigroup. [ABSTRACT FROM AUTHOR]

A fence is a particular partial order on a (finite) set, close to the linear order. In this paper, we calculate the rank of the semigroup FI n of all order-preserving partial injections on an n-element fence. In particular, we provide a minimal generating set for FI n . In the present paper, n is odd since this problem for even n was already solved by I. Dimitrova and J. Koppitz. [ABSTRACT FROM AUTHOR]

We say that an ultrafilter on an infinite group G is DTC if it determines the topological centre of the semigroup β G . If G has a subgroup of finite index in which conjugacy classes are all finite and uniformly bounded in size, then G does not admit a DTC ultrafilter. On the other hand, if G has no subgroup of finite index in which all conjugacy classes are finite, then G does admit a DTC ultrafilter. It follows that an infinite finitely generated group admits a DTC ultrafilter if and only if it has no abelian subgroup of finite index. [ABSTRACT FROM AUTHOR]

We study the relationship between the graph inverse semigroups of two graphs when there is a directed immersion between the graphs and we provide structural information about graph inverse semigroups of finite graphs that admit a directed cover onto a bouquet of circles. We provide a topological characterization of the universal groups of the local submonoids of a graph inverse semigroup. We also find necessary and sufficient conditions for a homomorphic image of a graph inverse semigroup to be another graph inverse semigroup. [ABSTRACT FROM AUTHOR]

Given a finite commutative semigroup (written additively), denote by [ABSTRACT FROM AUTHOR]

Given a ring R, we extend Ehrhard's linearization process by associating to any pre-finiteness space an R-module endowed with a Lefschetz topology. For a semigroup in the category of pre-finiteness spaces, one can endow this R-module with the convolution product to obtain an R-algebra. As examples of pre-finiteness spaces, we study topological spaces with bounded subsets (i.e., included in a compact) taken to be the finitary subsets. We prove that we obtain a finiteness space from any hemicompact space via this construction. As a corollary, any étale Hausdorff groupoid induces a semigroup in pre-finiteness spaces and its associated convolution algebra is complete in the hemicompact case. This is in particular the case for the infinite paths groupoid associated to any countable row-finite directed graph. [ABSTRACT FROM AUTHOR]

J. M. Howie has shown that the finite monogenic semigroups are absolutely closed. We provide a new, simple and direct proof of Howie's result. We also show that all the varieties defined by the identities ax = axa, axy = axay and axy = axyay are saturated if and only if they are epimorphically closed. [ABSTRACT FROM AUTHOR]

We give an efficient algorithm for the enumeration up to isomorphism of the inverse semigroups of order n, and we count the number S(n) of inverse semigroups of order n ≤ 15 . This improves considerably on the previous highest-known value S(9). We also give a related algorithm for the enumeration up to isomorphism of the finite inverse semigroups S with a given underlying semilattice of idempotents E, a given restriction of Green's D -relation on S to E, and a given list of maximal subgroups of S associated to the elements of E. [ABSTRACT FROM AUTHOR]

An inverse semigroup is finite if and only if it is finitely-generated, its singletons are recognizable, and the family of its recognizable subsets is closed under complex product. In particular, inverse Kleene semigroups are finite. [ABSTRACT FROM AUTHOR]

Given a vector space V and a subspace W of V, we consider the semigroup (under composition) S(V, W), consisting of all linear transformations on V which leave the subspace W invariant. In this paper, we characterize the natural partial order on S(V, W). In the partially ordered set, we determine the compatibility of their elements, and find all maximal and minimal elements. Also, we give necessary and sufficient conditions for S(V, W) to be factorizable, unit-regular and directly finite. [ABSTRACT FROM AUTHOR]

The purpose of this note is to prove that if a completely regular semigroup with finite H -classes posesses an idempotent-fixing automorphism of prime order, then it is an over-nilpotent semigroup. [ABSTRACT FROM AUTHOR]

We study growth of inverse semigroups defined by finite presentations. Let S be a finitely presented Rees quotient of a free inverse semigroup given by an irredundant presentation with n generators and m relators. We show that if S has polynomial growth, then m ≥ n 2 - 1 and this estimate is sharp. For any positive integer n, we also find, up to isomorphism, syntactic descriptions of all presentations that achieve this sharp lower bound. As part of the process, we describe all irredundant presentations of finite Rees quotients of free inverse semigroups having rank n, with the smallest number, namely n 2 , of relators. [ABSTRACT FROM AUTHOR]

A totally ordered monoid, or tomonoid for short, is a monoid endowed with a compatible total order. We reconsider in this paper the problem of describing the one-element Rees coextensions of a finite, negative tomonoid S, that is, those tomonoids that are by one element larger than S and whose Rees quotient by the poideal consisting of the two smallest elements is isomorphic to S. We show that any such coextension is a quotient of a pomonoid R (S) , called the free one-element Rees coextension of S. We investigate the structure of R (S) and describe the relevant congruences. We moreover introduce a finite family of finite quotients of R (S) from which the coextensions arise in a particularly simple way. [ABSTRACT FROM AUTHOR]

We present necessary conditions for the diagonal bi-act $$_S(S\times S)_S$$ over an epigroup $$S$$ to be finitely generated. They imply that this bi-act is not finitely generated whenever $$S$$ is infinite and locally finite. This solves a problem posed by Gallagher. [ABSTRACT FROM AUTHOR]

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Correction to: Semigroup Forum (2022) 104:166-179 https://doi.org/10.1007/s00233-021-10237-3 This article was updated to correct an error in the received date made during the production process. [Extracted from the article]

For any semigroup S a regular semigroup is described here that embeds S and is such that its non-trivial maximal subgroups are isomorphic to the Schützenberger groups of S, its Green’s relations restrict to the corresponding Green’s relations on S and it is finite when S is finite. [ABSTRACT FROM AUTHOR]

We investigate minimal, compact, square-closed subsets of the unit circle by identifying them with subsets of [0,1] and with sets of infinite words on two symbols; in particular, such sets arising from substitution maps. We consider problems on finiteness, recognizability, square roots, density and measure. Properties of the free semigroup on two symbols play a significant role in the analysis. [ABSTRACT FROM AUTHOR]