Finite Semigroups Research Articles

Products of quasi-idempotents in finite symmetric inverse semigroups

Let \(X_{n}=\{1,2,\ldots ,n\}\) and let \({\mathcal {S}}_{n}\) and \({\mathcal {I}}_{n}\) be the symmetric group and symmetric inverse semigroup on \(X_{n}\) respectively. In this paper, we show that the semigroup \({\mathcal {SI}}_{n}={\mathcal {I}}_{n}{\setminus } {\mathcal {S}}_{n}\), of all strictly partial one-to-one maps on \(X_{n}\), is generated by quasi-idempotent elements (non-idempotent elements \(\alpha \) satisfying \(\alpha ^{4}=\alpha ^{2}\)). Also, we give the least upper bound for the minimum length factorisation of each \(\alpha \in {\mathcal {SI}}_{n}\) into a product of quasi-idempotents.

Compact noncontraction semigroups of affine operators

We analyze compact multiplicative semigroups of affine operators acting in a finite-dimensional space. The main result states that every such semigroup is either contracting, that is, contains elements of arbitrarily small operator norm, or all its operators share a common invariant affine subspace on which this semigroup is contracting. The proof uses functional difference equations with contraction of the argument. We look at applications to self-affine partitions of convex sets, the investigation of finite affine semigroups and the proof of a criterion of primitivity for nonnegative matrix families. Bibliography: 32 titles.

Natural dualities, nilpotence and projective planes

We use an interpretation of projective planes to show the inherent nondualisability of some finite semigroups. The method is sufficiently flexible to demonstrate the nondualisability of (asymptotically) almost all finite semigroups as well as to give a fresh proof of the Quackenbush-Szabó result that any finite group with a nonabelian Sylow subgroup is nondualisable. A novel feature is that the ostensibly different notions of nilpotence for semigroups, nilpotence for groups, and the property of being nonorthodox for a completely 0-simple semigroup are unified by way of a single construction. We also give a semigroup example of two dualisable finite semigroups whose direct product is inherently nondualisable.

Finite involution semigroups with infinite irredundant bases of identities

Abstract A basis of identities for an algebra is irredundant if each of its proper subsets fails to be a basis for the algebra. The first known examples of finite involution semigroups with infinite irredundant bases are exhibited. These involution semigroups satisfy several counterintuitive properties: their semigroup reducts do not have irredundant bases, they share reducts with some other finitely based involution semigroups, and they are direct products of finitely based involution semigroups.

On residually finite semigroups of cellular automata

We prove that if M is a monoid and A a nite set with more than one element, then the residual niteness of M is equivalent to that of the monoid consisting of all cellular automata over M with alphabet A.

Semigroup graded algebras and codimension growth of graded polynomial identities

We show that if T is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative T-graded algebra over a field of characteristic 0 such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic 0 with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if T is a left or right zero band and the T-graded algebra is unital, or T is a cancellative semigroup, then the T-graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras.

Ample semigroups and Frobenius algebras

We prove that the semigroup algebra of an ample semigroup \(S\) over a field is Frobenius if and only if \(S\) is a finite inverse semigroup.

The commuting graph of the symmetric inverse semigroup

The commuting graph of a finite non-commutative semigroup $S$, denoted $\cg(S)$, is a simple graph whose vertices are the non-central elements of $S$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Let $\mi(X)$ be the symmetric inverse semigroup of partial injective transformations on a finite set $X$. The semigroup $\mi(X)$ has the symmetric group $\sym(X)$ of permutations on $X$ as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of $\sym(X)$. In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of $\cg(\sym(X))$, and in 2011, Dol\u{z}an and Oblak claimed (but their proof has a GAP) that this upper bound is in fact the exact value. The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup $\mi(X)$. We calculate the clique number of $\cg(\mi(X))$, the diameters of the commuting graphs of the proper ideals of $\mi(X)$, and the diameter of $\cg(\mi(X))$ when $|X|$ is even or a power of an odd prime. We show that when $|X|$ is odd and divisible by at least two primes, then the diameter of $\cg(\mi(X))$ is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of $\mi(X)$ of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of $\mi(X)$. The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.

Non-finitely based monoids

We present a general method for proving that a semigroup is non-finitely based (NFB). The method is strong enough to cover a majority of existing non-finite basis arguments for a periodic semigroups and allows to generalize several previous results and to simplify their proofs. The method also allows to remove one of the requirements on the “special system of identities” used by Perkins in 1968 to find the first two examples of finite NFB semigroups. We use our method to prove eleven new sufficient conditions under which a monoid is NFB. As an application, we find infinitely many new examples of finite finitely based aperiodic monoids whose direct product is NFB.

Davenport constant for semigroups II

Let S be a finite commutative semigroup. The Davenport constant of S, denoted D(S), is defined to be the least positive integer ℓ such that every sequence T of elements in S of length at least ℓ contains a proper subsequence T′ (T′≠T) with the sum of all terms from T′ equaling the sum of all terms from T. Let q>2 be a prime power, and let Fq[x] be the ring of polynomials over the finite field Fq. Let R be a quotient ring of Fq[x] with 0≠R≠Fq[x]. We prove thatD(SR)=D(U(SR)), where SR denotes the multiplicative semigroup of the ring R, and U(SR) denotes the group of units in SR.

Approximate biprojectivity of certain semigroup algebras

In this paper, we investigate the notion of approximate biprojectivity for semigroup algebras and for some Banach algebras related to semigroup algebras. We show that $$\ell ^{1}(S)$$ is approximately biprojective if and only if $$\ell ^{1}(S)$$ is biprojective, provided that $$S$$ is a uniformly locally finite inverse semigroup. Also for a Clifford semigroup $$S$$ , we show that approximate biprojectivity of $$\ell ^{1}(S)^{**}$$ gives pseudo-amenability of $$\ell ^{1}(S)$$ . We give a class of Banach algebras related to semigroup algebras which is not approximately biprojective.

QUASI-COMMUTATIVE SEMIGROUPS OF FINITE ORDER RELATED TO HAMILTONIAN GROUPS

If for every elements x and y of an associative algebraic structure (S, <TEX>${\cdot}$</TEX>) there exists a positive integer r such that <TEX>$ab=b^ra$</TEX>, then S is called quasi-commutative. Evidently, every abelian group or commutative semigroup is quasi-commutative. Also every finite Hamiltonian group that may be considered as a semigroup, is quasi-commutative however, there are quasi-commutative semigroups which are non-group and non commutative. In this paper, we provide three finitely presented non-commutative semigroups which are quasi-commutative. These are the first given concrete examples of finite semigroups of this type.

Fourier Inversion for Finite Inverse Semigroups

This paper continues the study of Fourier transforms on finite inverse semigroups, with a focus on Fourier inversion theorems and FFTs for new classes of inverse semigroups. We begin by introducing four inverse semigroup generalizations of the Fourier inversion theorem for finite groups. Next, we describe a general approach to the construction of fast inverse Fourier transforms for finite inverse semigroups complementary to an approach to FFTs given in previous work. Finally, we give fast inverse Fourier transforms for the symmetric inverse monoid and its wreath product by arbitrary finite groups, as well as fast Fourier and inverse Fourier transforms for the planar rook monoid, the partial cyclic shift monoid, and the partial rotation monoid.

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).

Certain Ranks of the Quotient Semigroup S/ρ and Prime Subsets of a Semigroup

In this paper we find a relationship between r2(S) and r2(S⁄ρ) where S is a finite semigroup, ρis a congruence on S and S /ρis the quotient semigroup (/S/ ≥ 2and/ S⁄ρ/ ≥ 2). We also determine r5(S⁄ρ) under certain conditions. Moreover we find prime subsets of S/ρ.

McCammond’s normal forms for free aperiodic semigroups revisited

AbstractThis paper revisits the solution of the word problem for ${\it\omega}$-terms interpreted over finite aperiodic semigroups, obtained by J. McCammond. The original proof of correctness of McCammond’s algorithm, based on normal forms for such terms, uses McCammond’s solution of the word problem for certain Burnside semigroups. In this paper, we establish a new, simpler, correctness proof of McCammond’s algorithm, based on properties of certain regular languages associated with the normal forms. This method leads to new applications.

Semigroup presentations for test local groups

In this paper we exhibit a type of semigroup presentations which determines a class of local groups. We show that the finite elements of this class generate the pseudovariety ${\bf LG}$ of all finite local groups and use them as test-semigroups to prove that ${\bf LG}$ and ${\bf S}$, the pseudovariety of all finite semigroups, verify the same $\kappa$-identities involving $\kappa$-terms of rank at most 1, where $\kappa$ denotes the implicit signature consisting of the multiplication and the $(\omega-1)$-power.

Automaton semigroup constructions

The aim of this paper is to investigate whether the class of automaton semigroups is closed under certain semigroup constructions. We prove that the free product of two automaton semigroups that contain left identities is again an automaton semigroup. We also show that the class of automaton semigroups is closed under the combined operation of 'free product followed by adjoining an identity'. We present an example of a free product of finite semigroups that we conjecture is not an automaton semigroup. Turning to wreath products, we consider two slight generalizations of the concept of an automaton semigroup, and show that a wreath product of an automaton monoid and a finite monoid arises as a generalized automaton semigroup in both senses. We also suggest a potential counterexample that would show that a wreath product of an automaton monoid and a finite monoid is not a necessarily an automaton monoid in the usual sense.

Partition monoids and embeddings in 2-generator regular $$*$$ ∗ -semigroups

The partition monoid \(\mathcal P_n\) is known to be minimally \(4\)-generated (for \(n\ge 3\)). Modulo some small values of \(n\), we show that: (1) \(\mathcal P_n\) embeds in a \(3\)-generator subsemigroup of \(\mathcal P_{n+1}\); (2) \(\mathcal P_n\) does not embed in a \(2\)-generator subsemigroup of \(\mathcal P_{n+1}\); and (3) \(\mathcal P_n\) embeds in a \(2\)-generator subsemigroup of \(\mathcal P_{n+3}\). A consequence of (3) is that every finite semigroup embeds in a finite \(2\)-generator regular \(*\)-semigroup.

Book review of “The $$q$$ q -theory of finite semigroups”

Book review of “The $$q$$ q -theory of finite semigroups”