Class Of Finite Semigroups Research Articles

Semigroups locally embeddable into the class of finite semigroups

In this paper the concept of local embeddability into finite structures (being LEF) is examined for the class of semigroups. The established results include the connections to the previously studied class of LEF groups, residual finiteness, linear semigroups and the preservation of being LEF under certain semigroup constructions, such as adjoining zero or identity and taking direct, semidirect and wreath products.

The Intersection Problem for Finite Semigroups

The intersection problem for finite semigroups asks, given a set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. In previous work, it was shown that is problem is [Formula: see text]-complete. We introduce compressibility measures as a useful tool to classify the complexity of the intersection problem for certain classes of finite semigroups. Using this framework, we obtain a new and simple proof that for groups and for commutative semigroups, the problem (as well as the variant where the languages are represented by finite automata) is contained in [Formula: see text]. We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove [Formula: see text]-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply decidability in poly-logarithmic time on alternating random-access Turing machines with a single alternation. We also establish connections to the monoid variant of the problem.

Commuting Regularity degree of finite semigroups

A pair (x, y) of elements x and y of a semigroup S is said to be a commuting regular pair, if there exists an element z ∈ S such that xy = (yx)z(yx). In a finite semigroup S, the probability that the pair (x, y) of elements of S is commuting regular will be denoted by dcr(S) and will be called the Commuting Regularity degree of S. Obviously if S is a group, then dcr(S) = 1. However for a semigroup S, getting an upper bound for dcr(S) will be of interest to study and to identify the different types of non-commutative semigroups. In this paper, we calculate this probability for certain classes of finite semigroups. In this study we managed to present an interesting class of semigroups where the probability is 1/2. This helps us to estimate a condition on non-commutative semigroups such that the commuting regularity of (x, y) yields the commuting regularity of (y, x).

ORDERED SEMIGROUPS, UPPER-TRIANGULAR REFLEXIVE RELATIONS AND SEMIGROUPS OF LANGUAGES

We study the connection between the classes of semigroups of positive languages, upper-triangular reflexive relations over posets and semigroups of order-preserving and extensive transformations of sets. From our constructions we derive some interesting consequences also for the finite case. For example, we prove that the class of finite semigroups of languages coincides with the class of semigroups of reflexive upper-triangular binary relations on finite sets.

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.

On Using the Join Operation to Define Classes of Algebras

We call a class of algebras a finitary prevariety if the class is closed under the formation of subalgebras and finitary direct products, and contains the one-element algebra. The join of two finitary prevarieties and a concept of a join-irreducible finitary prevariety may be introduced naturally. We develop techniques for proving that a finitary prevariety of semigroups is join-irreducible, and find many examples of join-irreducible finitary prevarieties of semigroups. For example, we prove that if a class of finite semigroups is defined by ω-identities and contains the class J, then it is a join-irreducible finitary prevariety.

Möbius functions and semigroup representation theory

This paper explores several applications of Möbius functions to the representation theory of finite semigroups. We extend Solomon's approach to the semigroup algebra of a finite semilattice via Möbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix algebras over group rings. We also extend work of Bidigare, Hanlon, Rockmore and Brown on calculating eigenvalues of random walks associated to certain classes of finite semigroups; again Möbius functions play an important role.

Finiteness Properties of Varieties and the Restriction to Finite Algebras

It is known that the variety generated by a finite semigroup is finitely axiomatisable if and only if it is finitely axiomatisable in the class of finite semigroups (M. Sapir). We examine similar restrictions for most other common finiteness properties of semigroup varieties.

Finite Proper covers in a class of finite semigroups with commuting idempotents

Weakly left ample semigroups are a class of semigroups that are (2,1)-subalgebras of semigroups of partial transformations, where the unary operation takes a transformation α to the identity map in the domain of α. It is known that there is a class of proper weakly left ample semigroups whose structure is determined by unipotent monoids acting on semilattices or categories. In this paper we show that for every finite weakly left ample semigroup S, there is a finite proper weakly left ample semigroup Ŝ and an onto morphism from Ŝ to S which separates idempotents. In fact, Ŝ is actually a (2,1)-subalgebra of a symmetric inverse semigroup, that is, it is a left ample semigroup (formerly, left type A).

Studying semigroups of mappings using quasi-identities

We define pq-varieties as classes of finite semigroups closed under the formation of subsemigroups and finitary direct products. Different examples of pq-varieties arise naturally in the study of semigroups of mappings. We consider for these pq-varieties two typical problems of the theory of pseudovarieties.

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties \(\mathfrak{A}_{\text{1}} \subset \mathfrak{B}_{\text{1}} \subset \mathfrak{A}_{\text{2}} \subset \mathfrak{B}_{\text{2}} \subset ...\) such that, for all i, an equational theory for \(\mathfrak{A}_i \) and for the class \(\mathfrak{A}_i \cap \mathfrak{F}\) of all finite semigroups in \(\mathfrak{A}_i \) is undecidable while an equational theory for \(\mathfrak{B}_i \) and for the class \(\mathfrak{B}_i \cap \mathfrak{F}\) of all finite semigroups in \(\mathfrak{B}_i \) is decidable. An infinite sequence of finitely based semigroup varieties \(\mathfrak{A}_{\text{1}} \supset \mathfrak{B}_i \supset \mathfrak{A}_{\text{2}} \supset \mathfrak{B}_{\text{2}} \supset ...\) is constructed so that, for all i, an equational theory for \(\mathfrak{B}_i \) and for the class \(\mathfrak{B}_i \cap \mathfrak{F}\) of all finite semigroups in \(\mathfrak{B}_i \) is decidable whicle an equational theory for \(\mathfrak{A}_i \) and for the class \(\mathfrak{A}_i \cap \mathfrak{F}\) of all finite semigroups in \(\mathfrak{A}_i \) is not.

The embeddability of ring and semigroup amalgams is undecidable

AbstractWe prove that there is no algorithm to determine when an amalgam of finite rings (or semigroups) can be embedded in the class of rings or in the class of finite rings (respectively, in the class of semigroups or in the class of finite semigroups). These results are in marked contrast with the corresponding problems for groups where every amalgam of finite groups can be embedded in a finite group.

Small semigroup related structures with infinite properties

In mathematics, one frequently encounters constructions of a pathological or critical nature. In this thesis we investigate such structures in semigroup theory with a particular aim of finding small, finite, examples with certain associated infinite characteristics. We begin our investigation with a study of the identities of finite semigroups. A semigroup (or the variety it generates) whose identities admit a finite basis is said to be finitely based. We find examples of pairs of finite (aperiodic) finitely based semigroups whose direct product is not finitely based (answering a question of M. Sapir) and of pairs of finite (aperiodic) semigroups that are not finitely based whose direct product is finitely based. These and other semigroups from a large class (the class of finite Rees quotients of free monoids) are also shown to generate varieties with a chain of finitely generated supervarieties which alternate between being finitely based and not finitely based. Furthermore it is shown that in a natural sense, almost semigroups from this class are not finitely based. Not finitely based semigroups that are locally finite and have the property that every locally finite variety containing them is also not finitely based are said to be inherently not finitely based. We construct all minimal inherently not finitely based divisors in the class of finite semigroups and establish several results concerning a fundamental example with this property; the six element Brandt semigroup with adjoined identity element, B. 1/2. We then find the first examples of finite semigroups admitting a finite basis of identities but generating a variety with uncountably many subvarieties (indeed with a chain of subvarieties with the same ordering as the real numbers). For some well known classes, a complete description of the members with this property are obtained and related examples and results concerning joins of varieties are also found. A connection between these results and the construction of varieties with decidable word problem but undecidable uniform word problem is investigated. Finally we investigate several embedding problems not directly concerned with semigroup varieties and show that they are undecidable. The first and second of these problems concern the fundamental relations of Green; in addition some small examples are found which exhibit unusual related properties and a problem of M. Sapir is solved. The third of the embedding problems concerns the potential embeddability of finite semigroup amalgams. The results are easily extended to the class of rings.

Some undecidable embedding problems for finite semigroups

Let S be a finite semigroup, A be a given subset of S and L, R, H, D and J be Green's equivalence relations. The problem of determining whether there exists a supersemigroup T of S from the class of all semigroups or from the class of finite semigroups, such that A lies in an L or R-class of T has a simple and well known solution (see for example [7], [8] or [3]). The analogous problem for J or D relations is trivial if T is of arbitrary size, but undecidable if T is required to be finite [4] (even if we restrict ourselves to the case |A| = 2 [6]). We show that for the relation H, the corresponding problem is undecidable in both the class of finite semigroups (answering Problem 1 of [9]) and in the class of all semigroups, extending related results obtained by M. V. Sapir in [9]. An infinite semigroup with a subset never lying in a H-class of any embedding semigroup is known and, in [9], the existence of a finite semigroup with this property is established. We present two eight element examples of such semigroups as well as other examples satisfying related properties.

FINITE FILTERING SEMIGROUPS

A semigroup is called filtering if each of its subsemigroups has the smallest (with respect to inclusion) generating set. It is proved in this article that every maximal chain of nonempty subsemigroups of a finite filtering semigroup has length equal to the order of the semigroup, and that filtering semigroups are characterized by this property in the class of finite semigroups. The main result is a characterization of the class of finite filtering semigroups by means of forbidden divisors, to which end the author finds all finite nonfiltering semigroups all of whose proper divisors are filtering semigroups.

On the linear orderability of two classes of finite semigroups

On the linear orderability of two classes of finite semigroups

On a problem of M. P. Schützenberger

A class of finite semigroups is called a genus if it is closed under homomorphic images, subsemigroups and finite direct products. During a talk at the Symposium on Semigroups held at the University of St Andrews, in 1976, M. P. Schützenberger posed the problem of characterising the smallest genus which contains finite groups and finite semigroups, all of whose subgroups are trivial.