Finite Semigroups Research Articles

Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups

Let TX be the full transformation monoid over a finite set X, and fix some a∈TX of rank r. The variant TXa has underlying set TX, and operation f⋆g=fag. We study the congruences of the subsemigroup P=Reg(TXa) consisting of all regular elements of TXa, and the lattice Cong(P) of all such congruences. Our main structure theorem ultimately decomposes Cong(P) as a specific subdirect product of Cong(Tr), and the full equivalence relation lattices of certain combinatorial systems of subsets and partitions. We use this to give an explicit classification of the congruences themselves, and we also give a formula for the height of the lattice.

Strongly nonfinitely based monoids

We show that the 42 -element monoid of all partial order preserving and extensive injections on the 4 -element chain is not contained in any variety generated by a finitely based finite semigroup.

Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups

Given any quasi-countable, in particular, any countable inverse semigroup S , we introduce a way to equip S with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on S . This allows us to unambiguously define the uniform Roe algebra of S , which we prove can be realized as a canonical crossed product of \ell^{\infty}(S) and S . We relate these metrics to the analogous metrics on Hausdorff étale groupoids. Using this setting, we study those inverse semigroups with asymptotic dimension 0 . Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by S being locally \mathcal{L} -finite, and equivalently sparse as a metric space.

Heights of posets associated with Green's relations on semigroups

Given a semigroup S, for each Green's relation K∈{L,R,J,H} on S, the K-height of S, denoted by HK(S), is the height of the poset of K-classes of S. More precisely, if there is a finite bound on the sizes of chains of K-classes of S, then HK(S) is defined as the maximum size of such a chain; otherwise, we say that S has infinite K-height. We discuss the relationships between these four K-heights. The main results concern the class of stable semigroups, which includes all finite semigroups. In particular, we prove that a stable semigroup has finite L-height if and only if it has finite R-height if and only if it has finite J-height. In fact, for a stable semigroup S, if HL(S)=n then HR(S)≤2n−1 and HJ(S)≤2n−1, and we exhibit a family of examples to prove that these bounds are sharp. Furthermore, we prove that if 2≤HL(S)<∞ and 2≤HR(S)<∞, then HJ(S)≤HL(S)+HR(S)−2. We also show that for each n∈N there exists a semigroup S such that HL(S)=HR(S)=2n+n−3 and HJ(S)=2n+1−4. By way of contrast, we prove that for a regular semigroup the L-, R- and H-heights coincide with each other, and are greater or equal to the J-height. Moreover, in a stable, regular semigroup the L-, R-, H- and J-heights are all equal.

The number of spanning trees of E-extended power graphs associated with finite semigroups

We introduce a new kind of graph built from finite semigroups that we refer to as E-extended power graphs. An E-extended power graph ΓS of a finite semigroup S can be considered as the union of the power graph P(S) of S and the tree generated by E(S), where E(S) is the set of idempotents of S. Actually, ΓS is a minimal connected graph with vertex set S and containing P(S). In particular, if S is a finite semigroup with a unique idempotent we have ΓS=P(S). We give the number of spanning trees of an E-extended power graph of a finite semigroup, in particular, (Zn,⋅). Finally, as an application we discuss the number of spanning trees of an E-extended power graph of completely simple semigroups and completely regular semigroups.

On the Ryll-Nardzewski fixed point property in convex uniform spaces

In this paper, we introduce the concept of a convex uniform space as a natural generalization of locally convex spaces, and a fixed point property as a semigroup formulation of the famous Ryll-Nardzewski fixed point theorem, and study its non-linear counterpart in this framework. The Nardzewski's fixed point theorem is a beautiful result in fixed point theory that ensures the existence of a common fixed point for any noncontracting semigroup of continuous affine self-mappings on a non-void weakly compact convex set in a separated locally convex space. We establish that in a separated convex uniform space satisfying a certain property (P), any noncontracting finite semigroup of contraction mappings of a non-empty bounded closed convex subset possesses a common fixed point. In case where the underlying space is admissible, we prove that this result is extendable to arbitrary noncontracting semigroups without condition (P) if we assume that the phase space is weakly compact and convex.

The determinant of semigroups of the pseudovariety ECOM

The purpose of this paper is to compute the nonzero semigroup determinant of the class of finite semigroups in which every two idempotents commute. This class strictly contains the class of finite semigroups that have central idempotents and the class of finite inverse semigroups. This computation holds significance in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.

Finite regular semigroups with permutations that map elements to inverses

We give an account on what is known on the subject of permutation matchings, which are bijections of a finite regular semigroup that map each element to one of its inverses. This includes partial solutions to some open questions, including a related novel combinatorial problem.

Restrictions on local embeddability into finite semigroups

We expand the concept of local embeddability into finite structures (LEF) for the class of semigroups with investigations of non-LEF structures, a closely related generalising property of local wrapping of finite structures (LWF) and inverse semigroups. The established results include a description of a family of non-LEF semigroups unifying the bicyclic monoid and Baumslag–Solitar groups and demonstrating that inverse LWF semigroups with finite number of idempotents are LEF.

Characterization of Ordered Semigroups Generating Well Quasi-Orders of Words

The notion of a quasi-order generated by a homomorphism from the semigroup of all words onto a finite ordered semigroup was introduced by Bucher et al. (Theor. Comput. Sci. 40, 131–148 1985). It naturally occurred in their studies of derivation relations associated with a given set of context-free rules, and they asked a crucial question, whether the resulting relation is a well quasi-order. We answer this question in the case of the quasi-order generated by a semigroup homomorphism. We show that the answer does not depend on the homomorphism, but it is a property of its image. Moreover, we give an algebraic characterization of those finite semigroups for which we get well quasi-orders. This characterization completes the structural characterization given by Kunc (Theor. Comput. Sci. 348, 277–293 2005) in the case of semigroups ordered by equality. Compared with Kunc’s characterization, the new one has no structural meaning, and we explain why that is so. In addition, we prove that the new condition is testable in polynomial time.

Finite semigroups and periodic sums systems in βN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\mathbb {N}$$\\end{document} and their Ramsey theoretic consequences

Let m,n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m,n\\ge 2$$\\end{document} and define ν:ω→{0,…,m-1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u :\\omega \\rightarrow \\{0,\\ldots ,m-1\\}$$\\end{document} by ν(k)≡k(modm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u (k)\\equiv k\\pmod {m}$$\\end{document}. We construct some new finite semigroups in βN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\mathbb {N}$$\\end{document}, in particular, a semigroup generated by m elements of order n with cardinality mn+mn-1+⋯+m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m^n+m^{n-1}+\\cdots +m$$\\end{document}. We also show that, for n≥m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge m$$\\end{document}, there is a sequence p0,…,pm-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_0,\\ldots ,p_{m-1}$$\\end{document} in βN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\mathbb {N}$$\\end{document} such that all sums ∑j=ii+kpν(j)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{j=i}^{i+k}p_{\ u (j)}$$\\end{document}, where i∈{0,…,m-1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i\\in \\{0,\\ldots ,m-1\\}$$\\end{document} and k∈{0,…,n-1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in \\{0,\\ldots ,n-1\\}$$\\end{document}, are distinct and ∑j=ii+npν(j)=∑j=ii+n-mpν(j)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{j=i}^{i+n}p_{\ u (j)}=\\sum _{j=i}^{i+n-m}p_{\ u (j)}$$\\end{document} for each i. As consequences we derive some new Ramsey theoretic results. In particular, we show that, for n≥m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge m$$\\end{document}, there is a partition {Ai,k:(i,k)∈{0,…,m-1}×{0,…,n-1}}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{A_{i,k}:(i,k)\\in \\{0,\\ldots ,m-1\\}\ imes \\{0,\\ldots ,n-1\\}\\}$$\\end{document} of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {N}$$\\end{document} such that, whenever for each (i, k), Bi,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {B}_{i,k}$$\\end{document} is a finite partition of Ai,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{i,k}$$\\end{document}, there exist Bi,k∈Bi,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{i,k}\\in \\mathscr {B}_{i,k}$$\\end{document} and a sequence (xj)j=0∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x_j)_{j=0}^\\infty $$\\end{document} such that for every finite sequence j0<…<js\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j_0<\\ldots <j_s$$\\end{document} such that jt+1≡jt+1(modm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j_{t+1}\\equiv j_t+1\\pmod {m}$$\\end{document} for each t<s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t<s$$\\end{document}, one has xj0+⋯+xjs∈Bi0,k0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_{j_0}+\\cdots +x_{j_s}\\in B_{i_0,k_0}$$\\end{document}, where i0=ν(j0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i_0=\ u (j_0)$$\\end{document} and k0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_0$$\\end{document} is s if s≤n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\le n-1$$\\end{document} and n-m+ν(s-n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-m+\ u (s-n)$$\\end{document} otherwise.

Complemented zero-divisor graphs associated with finite commutative semigroups

Let S be a commutative semigroup with zero. The zero-divisor graph associated with S is the simple graph whose vertex set is given by the nonzero zero-divisors of S where two distinct vertices are adjacent if and only if their product is zero. A graph is called complemented if every vertex has an incident edge which is not the edge of any three cycle. Complemented zero-divisor graphs which are associated with a finite commutative semigroup are described in terms of their clique number. The algebraic properties of the finite semigroups S whose zero-divisor graphs are complemented are described in terms of the clique number of the graph. Infinite families of complemented zero-divisor graphs associated with a finite commutative semigroup are given.

Not all nilpotent monoids are finitely related

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

Semiring and involution identities of powers of inverse semigroups

The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis.

Embedding finite involution semigroups in matrices with transposition

The present article establishes two contrasting results: a finite involution semigroup is embeddable in some semigroup of complex matrices with conjugate transposition if and only if it is an inverse semigroup, while every finite involution semigroup is embeddable in some semigroup of binary matrices with the skew transposition across the secondary diagonal.

On the minimal faithful degree of Rhodes semisimple semigroups

In this paper we compute the minimal degree of a faithful representation by partial transformations of a finite semigroup admitting a faithful completely reducible matrix representation over the field of complex numbers. This includes all inverse semigroups, and hence our results generalize earlier results of Easdown and Schein on the minimal faithful degree of an inverse semigroup. It also includes well-studied monoids like full matrix monoids over finite fields and the monoid of binary relations (i.e., matrices over the Boolean semiring). Our answer reduces the computation to considerations of permutation representations of maximal subgroups that are faithful when restricted to distinguished normal subgroups. This is analogous to (and inspired by) recent results of the second author on the minimal number of irreducible constituents in a faithful completely reducible complex matrix representation of a finite semigroup that admits one. To illustrate what happens when a finite semigroup does not admit a faithful completely reducible representation, we compute the minimal faithful degree of the opposite monoid of the full transformation monoid.

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.

Regular semigroups weakly generated by idempotents

A regular semigroup is weakly generated by a set [Formula: see text] if it has no proper regular subsemigroup containing [Formula: see text]. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup [Formula: see text] weakly generated by [Formula: see text] idempotents such that all other regular semigroups weakly generated by [Formula: see text] idempotents are homomorphic images of [Formula: see text]. The semigroup [Formula: see text] is defined by a presentation [Formula: see text] and its structure is studied. Although each of the sets [Formula: see text], [Formula: see text], and [Formula: see text] is infinite for [Formula: see text], we show that the word problem is decidable as each congruence class has a “canonical form”. If [Formula: see text] denotes [Formula: see text] for [Formula: see text], we prove also that [Formula: see text] contains copies of all [Formula: see text] as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide [Formula: see text]; and (ii) all finite semigroups divide [Formula: see text].

Catalan monoids inherently nonfinitely based relative to finite [formula omitted]-trivial semigroups

We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite R-trivial semigroup. This provides unified proofs for several known facts and leads to a bunch of new results on the Finite Basis Problem for finite R- and J-trivial semigroups.

An algebraic characterization of self-generating chemical reaction networks using semigroup models

The ability of a chemical reaction network to generate itself by catalyzed reactions from constantly present environmental food sources is considered a fundamental property in origin-of-life research. Based on Kaufmann’s autocatalytic sets, Hordijk and Steel have constructed the versatile formalism of catalytic reaction systems (CRS) to model and to analyze such self-generating networks, which they named reflexively autocatalytic and food-generated. Recently, it was established that the subsequent and simultaenous catalytic functions of the chemicals of a CRS give rise to an algebraic structure, termed a semigroup model. The semigroup model allows to naturally consider the function of any subset of chemicals on the whole CRS. This gives rise to a generative dynamics by iteratively applying the function of a subset to the externally supplied food set. The fixed point of this dynamics yields the maximal self-generating set of chemicals. Moreover, the set of all functionally closed self-generating sets of chemicals is discussed and a structure theorem for this set is proven. It is also shown that a CRS which contains self-generating sets of chemicals cannot have a nilpotent semigroup model and thus a useful link to the combinatorial theory of finite semigroups is established. The main technical tool introduced and utilized in this work is the representation of the semigroup elements as decorated rooted trees, allowing to translate the generation of chemicals from a given set of resources into the semigroup language.