Finite Semigroups Research Articles

Involution matchings, the semigroup of orientation-preserving and orientation-reversing mappings, and inverse covers of the full transformation semigroup

We continue the study of permutations of a finite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We show, in two ways, how to construct an involution matching on the semigroup of all transformations which either preserve or reverse orientation of a cycle. Finally, as an application, we use involution matchings to prove that when the base set has at least four members, a finite full transformation semigroup has no cover by inverse subsemigroups that is closed under intersection.

Pointlike sets for varieties determined by groups

For a variety of finite groups H, let H‾ denote the variety of finite semigroups all of whose subgroups lie in H. We give a characterization of the subsets of a finite semigroup that are pointlike with respect to H‾. Our characterization is effective whenever H has a decidable membership problem. In particular, the separation problem for H‾-languages is decidable for any decidable variety of finite groups H. This generalizes Henckell's theorem on decidability of aperiodic pointlikes.

Pseudovarieties of Ordered Completely Regular Semigroups

This paper is a contribution to the theory of finite semigroups and their classification in pseudovarieties, which is motivated by its connections with computer science. The question addressed is what role can play the consideration of an order compatible with the semigroup operation. In the case of unions of groups, so-called completely regular semigroups, the problem of which new pseudovarieties appear in the ordered context is solved. As applications, it is shown that the lattice of pseudovarieties of ordered completely regular semigroups is modular and that taking the intersection with the pseudovariety of bands defines a complete endomorphism of the lattice of all pseudovarieties of ordered semigroups.

Unified theory for finite Markov chains

We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup S. Our methods use geometric finite semigroup theory via the Karnofsky–Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to P(n), the set of all subsets of an n element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or R-trivial bands) by Brown and Diaconis, extensions to R-trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The Karnofsky–Rhodes expansion of the right Cayley graph of S in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded S. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded S and the original S. Analyzing the normal forms also provides an estimate on the mixing time.

ON THE NUMBER OF SUBSEMIGROUPS OF DIRECT PRODUCTS INVOLVING THE FREE MONOGENIC SEMIGROUP

The direct product$\mathbb{N}\times \mathbb{N}$of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for$\mathbb{N}\times S$, where$S$is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if$S$is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of$S$has a relative left or right identity element.

Linear Conjugacy

AbstractWe say that two elements of a group or semigroup are $\Bbbk$-linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$-linear conjugacy for finite semigroups (and, in particular, for finite groups) over an arbitrary field $\Bbbk$.

Merge Decompositions, Two-sided Krohn–Rhodes, and Aperiodic Pointlikes

AbstractThis paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell’s aperiodic pointlike theorem. We use a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided semidirect product whose components are built from two subsemigroups $T_{1}$, $T_{2}$, which together generate $T$, and the subsemigroup generated by their setwise product $T_{1}T_{2}$. In this sense we decompose $T$ by merging the subsemigroups $T_{1}$ and $T_{2}$. More generally, our technique merges semigroup homomorphisms from free semigroups.

On the algebraic and arithmetic structure of the monoid of product-one sequences II

Let $G$ be a finite group and $G'$ its commutator subgroup. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The monoid $\mathcal B (G)$ of all product-one sequences over $G$ is a finitely generated C-monoid whence it has a finite commutative class semigroup. It is well-known that the class semigroup is a group if and only if $G$ is abelian (equivalently, $\mathcal B (G)$ is Krull). In the present paper we show that the class semigroup is Clifford (i.e., a union of groups) if and only if $|G'| \le 2$ if and only if $\mathcal B (G)$ is seminormal, and we study sets of lengths in $\mathcal B (G)$.

Finite Structurally Uniform Groups and Commutative Nilsemigroups

Let S be a finite semigroup. By Sub(S) we denote the lattice of all its subsemigroups. If A ∈ Sub(S), then by h(A) we denote the height of a subsemigroup A in the lattice Sub(S). A semigroup S is called structurally uniform if, for any A,B ∈ Sub(S) the condition h(A) = h(B) implies that A ≅ B. We present a classification of finite structurally uniform groups and commutative nilsemigroups.

Topologically congruence-free compact semigroups

A classical result of semigroup theory says that any finite congruence-free semigroup S (i.e., S has exactly two congruences) without zero such that card(S)>2 is a simple group. We shall show that an analogous result holds for any infinite topologically congruence-free compact semigroup (a compact semigroup A is topologically congruence-free if the set of its algebraic congruences ρ for which A/ρ is a compact semigroup, is equal to {1A,A×A}). In fact, every such semigroup must be a metric Lie group with cardinality c. Also, we prove that all topologically congruence-free compact semigroups with zero are (unfortunately) finite.

Nilpotency and strong nilpotency for finite semigroups

AbstractNilpotent semigroups in the sense of Mal’cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, MN, which has finite rank. The semigroup identities that define nilpotent semigroups lead us to define strongly Mal’cev nilpotent semigroups. Finite strongly Mal’cev nilpotent semigroups constitute a non-finite rank pseudovariety, SMN. The pseudovariety SMN is strictly contained in the pseudovariety MN, but all finite nilpotent groups are in SMN. We show that the pseudovariety MN is the intersection of the pseudovariety BGnil with a pseudovariety defined by a κ-identity. We further compare the pseudovarieties MN and SMN with the Mal’cev product 𝖩ⓜ𝖦nill.

The complexity of weakly recognizing morphisms

Weakly recognizing morphisms from free semigroups onto finite semigroups are a classical way for defining the class of ω-regular languages, i.e., a set of infinite words is weakly recognizable by such a morphism if and only if it is accepted by some Büchi automaton. We study the descriptional complexity of various constructions and the computational complexity of various decision problems for weakly recognizing morphisms. The constructions we consider are the conversion from and to Büchi automata, the conversion into strongly recognizing morphisms, as well as complementation. We also show that the fixed membership problem is NC1-complete, the general membership problem is in L and that the inclusion, equivalence and universality problems are NL-complete. The emptiness problem is shown to be NL-complete if the input is given as a non-surjective morphism.

Locally maximal regular subsemibands of the finite transformation semigroups $${\mathcal {T}}(n,r)$$ T ( n , r )

Let $${\mathcal {T}}_n$$ be the semigroup of all full transformations on the finite set $$X_n=\{1,2,\ldots , n\}$$ . For $$1 \le r \le n$$ , set $${\mathcal {T}}(n, r)=\{\alpha \in {\mathcal {T}}_n: {\mathrm {rank}}(\alpha )\le r\}$$ . In this paper we describe the locally maximal regular subsemibands of the finite transformation semigroups $${\mathcal {T}}(n,r)$$ and completely obtain their classification.

On a Java library to perform S-expansions of Lie algebras

The S-expansion method is a generalization of the Inönü-Wigner (IW) contraction that allows to study new non-trivial relations between different Lie algebras. Basically, this method combines a Lie algebra G with a finite abelian semigroup S in such a way that a new S-expanded algebra GS can be defined. When the semigroup has a zero-element and/or a specific decomposition, which is said to be resonant with the subspace structure of the original algebra, then it is possible to extract smaller algebras from GS which have interesting properties. Here we give a brief description of the S-expansion, its applications and the main motivations that lead us to elaborate a Java library, which automatizes this method and allows us to represent and to classify all possible S-expansions of a given Lie algebra.

Automorphisms of semigroups of k-linked upfamilies

A family $$ \mathcal{A} $$ of non-empty subsets of a set X is called an upfamily, if, for each set $$ A\in \mathcal{A} $$ ; any set B ⊃ A belongs to $$ \mathcal{A} $$ . An upfamily $$ \mathrm{\mathcal{L}} $$ is called k-linked, if $$ \cap \mathrm{\mathcal{F}}\ne \varnothing $$ for any subfamily $$ \mathrm{\mathcal{F}}\subset \mathrm{\mathcal{L}} $$ of cardinality $$ \left|\mathrm{\mathcal{F}}\right|\le k $$ . The extension Nk(X) consists of all k-linked upfamilies on X. Any associative binary operation ∗ : X × X → X can be extended to an associative binary operation ∗ : Nk(X) × Nk(X) → Nk(X). Here, we study automorphisms of the extensions of groups and finite monogenic semigroups. We also describe the automorphism groups of extensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups.

Algebras of right ample semigroups

AbstractStrict RA semigroups are common generalizations of ample semigroups and inverse semigroups. The aim of this paper is to study algebras of strict RA semigroups. It is proved that any algebra of strict RA semigroups with finite idempotents has a generalized matrix representation whose degree is equal to the number of non-zero regular 𝓓-classes. In particular, it is proved that any algebra of finite right ample semigroups has a generalized upper triangular matrix representation whose degree is equal to the number of non-zero regular 𝓓-classes. As its application, we determine when an algebra of strict RA semigroups (right ample monoids) is semiprimitive. Moreover, we prove that an algebra of strict RA semigroups (right ample monoids) is left self-injective iff it is right self-injective, iff it is Frobenius, and iff the semigroup is a finite inverse semigroup.

Lie-Markov Models Derived from Finite Semigroups.

We present and explore a general method for deriving a Lie-Markov model from a finite semigroup. If the degree of the semigroup is k, the resulting model is a continuous-time Markov chain on k-states and, as a consequence of the product rule in the semigroup, satisfies the property of multiplicative closure. This means that the product of any two probability substitution matrices taken from the model produces another substitution matrix also in the model. We show that our construction is a natural generalization of the concept of group-based models.

Variety Membership Problem for Two Classes of Non-Finitely Based Semigroups

The variety membership problem for two classes of non-finitely based semigroups is considered. It is shown that a finite semigroup S belongs to the variety generated by one of these non-finitely based semigroups if and only if S satisfies four certain equations that involve at most 2│S│+1 distinct variables.

Varieties of Boolean inverse semigroups

In an earlier work, the author observed that Boolean inverse semigroups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of varieties of biases in terms of varieties of groups:(1)Every free bias is residually finite. In particular, the word problem for free biases is decidable.(2)Every proper variety of biases contains a largest finite symmetric inverse semigroup, and it is generated by its members that are monoids of generalized rook matrices over groups with zero.(3)There is an order-preserving, one-to-one correspondence between proper varieties of biases and certain finite sequences of varieties of groups, descending in a strong sense defined in terms of wreath products by finite symmetric groups.

On Vertex-transitive Cayley Graphs of Finite Transformation Semigroups with Restricted Range

Let \(T(X)\) be the semigroup of all transformations on a set \(X\). For a non-empty subset \(Y\) of \(X\), denoted by \(T(X,Y)\) the subsemigroup of \(T(X)\) consisting of all transformations whose range is contained in \(Y\). Kelarev and Praeger in [9] gave necessary and sufficient conditions for all vertex-transitive Cayley graphs of semigroups. In this paper, we give similar descriptions for all vertex-transitive Cayley graphs of \(T(X,Y)\).