Finite Semigroups Research Articles

О ГЕНЕРИЧЕСКОЙ СЛОЖНОСТИ ПРОБЛЕМЫ ИЗОМОРФИЗМА КОНЕЧНЫХ ПОЛУГРУПП

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, V. Kapovich, P. Schupp, and V. Shpilrain in 2003. This approach studies behavior of an algorithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we study the generic complexity of the isomorphism problem for finite semigroups. In this problem, for any two semigroups of the same order, given by their multiplication tables, it is required to determine whether they are isomorphic. V. Zemlyachenko, N. Korneenko, and R. Tyshkevich in 1982 proved that the graph isomorphism problem polynomially reduces to this problem. The graph isomorphism problem is a well-known algorithmic problem that has been actively studied since the 1970s, and for which polynomial algorithms are still unknown. So from a computational point of view the studied problem is no simpler than the graph isomorphism problem. We present a generic polynomial algorithm for the isomorphism problem of finite semigroups. It is based on the characterization of almost all finite semigroups as 3-nilpotent semigroups of a special form, established by D. Kleitman, B. Rothschild, and J. Spencer, as well as the Bollobas polynomial algorithm, which solves the isomorphism problem for almost all strongly sparse graphs.

Multiplicative invertibility characterization on star-like cyclicpoid CyPω*n finite partial transformation semigroups

Multiplicative invertibility characterization on star-like cyclicpoid CyPω*n finite partial transformation semigroups

SOME IMPORTANT APPLICATIONS OF SEMIGROUPS

This Paper deals with the some important applications of semigroups in general and regular semigroups in particular.The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid. In probability theory, semigroups are associated with Markov process. In section 2 we have seen different areas of applications of semigroups. We identified some Applications in biology, Partial Differential equation, Formal Languages etc whose semigroup structures are nothing but regular.

EQUATIONS OVER DIRECT POWERS OF ALGEBRAIC STRUCTURES IN RELATIONAL LANGUAGES

For a semigroup S (group G) we study relational equations and describe all semigroups S with equationally Noetherian direct powers. It follows that any group G has equationally Noetherian direct powers if we consider G as an algebraic structure of a certain relational language. Further we specify the results as follows: if a direct power of a finite semigroup S is equationally Noetherian, then the minimal ideal Ker(S) of S is a rectangular band of groups and Ker(S) coincides with the set of all reducible elements

About the class of approximation by generalized characters

Given a certain class K1 of algebraic structures. We study a problem of finding a class K2 of algebraic structures such that the class K1 is approximable into K2 with respect to various predicates by generalized characters from K1 to K2. The problem of minimization of approximation is also considered. Some theorems related to the problem of constructing an approximation class are obtained. The problem in question is much more complicated and actual than the approximation problem we have been studying before (see [2]-[6]). The results of the description of the approximation class play an important role in studying the solvability problem of the predicate P in the class of semigroups K. In particular, if the approximation class consists of finite semigroups, then this problem is solved positively. Even more difficult is the problem of determining the necessary conditions that class is an approximation class for a given class K.

Characterizations of Cayley graphs of finite transformation semigroups with restricted range

The transformation semigroup with restricted range [Formula: see text] is the set of all functions from a set [Formula: see text] into a non-empty subset [Formula: see text] of [Formula: see text]. In this paper, we characterize Cayley graphs of [Formula: see text] with the connection set [Formula: see text]. Moreover, the undirected property of Cayley graphs Cay[Formula: see text] is studied.

Infinite Dimensional Systems of Particles with Interactions Given by Dunkl Operators

Firstly we consider a finite dimensional Markov semigroup generated by Dunkl Laplacian with drift terms. For this semigroup we prove gradient bounds involving a symmetrised carre-du-champ operator, and we show that for small coefficients this semigroup has a unique invariant measure which satisfies ergodicity properties. We then extend this analysis to an infinite dimensional model on $(\mathbb {R}^{N})^{\mathbb {Z}^{d}}$ , consisting of interacting finite dimensional models. We construct an associated Markov semigroup for this model using gradient bounds, and finally we study the existence of invariant measures and ergodicity properties.

Post theorem for strongly dependent n-ary semigroups

Abstract We use an analogue of Gluskin–Hosszú theorem for strongly dependent finite n-ary semigroups to prove an analogue of Post coset theorem. We show that in fact these theorems are equivalent. Thus it turns our that the case of n-ary semigroups is fully similar to the case of n-groups.

On the decoupled Markov group conjecture

The Markov group conjecture, a long-standing open problem in the theory of Markov processes with countable state space, asserts that a strongly continuous Markov semigroup $T = (T_t)_{t \in [0,\infty)}$ on $\ell^1$ has bounded generator if the operator $T_1$ is bijective. Attempts to disprove the conjecture have often aimed at glueing together finite dimensional matrix semigroups of growing dimension - i.e., it was tried to show that the Markov group conjecture is false even for Markov processes that decouple into (infinitely many) finite dimensional systems. In this article we show that such attempts must necessarily fail, i.e., we prove the Markov group conjecture for processes that decouple in the way described above. In fact, we even show a more general result that gives a universal norm estimate for bounded generators $Q$ of positive semigroups on any Banach lattice. Our proof is based on a filter product technique, infinite dimensional Perron-Frobenius theory and Gelfand's $T = \operatorname{id}$ theorem.

Structure of long idempotent-sum-free sequences over finite cyclic semigroups

Let [Formula: see text] be a finite cyclic semigroup written additively. An element [Formula: see text] of [Formula: see text] is said to be idempotent if [Formula: see text]. A sequence [Formula: see text] over [Formula: see text] is called idempotent-sum-free provided that no idempotent of [Formula: see text] can be represented as a sum of one or more terms from [Formula: see text]. We prove that an idempotent-sum-free sequence over [Formula: see text] of length over approximately a half of the size of [Formula: see text] is well structured. This result generalizes the Savchev–Chen Structure Theorem for zero-sum-free sequences over finite cyclic groups.

Numerical Solutions of the Work Done on Finite Order-Preserving Injective Partial Transformation Semigroup

In this research work, we have used an effective methodology to obtain formulas for calculating the total work done and consequently the average work done and the power of the transformation by elements of finite order-preserving injective partial transformation semigroup. The generalized formulas was applied to obtain the numerical solutions and results tabulated. We equally plot graphs to illustrate the nature of the total work done and the average work done by elements of the finite order-preserving partial transformation semigroup. Results obtained showed that moving the elements from domain to co-domain involve allot of work to be done on a numerical scale

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.

Classes of non-commutative semigroups with finite Fibonacci invariants

For a $2$-generated semigroup $S$ we define the invariant $\lambda (S)$ as the minimum of the Fibonacci lengths over all generating pairs of the semigroup $S$, where the Fibonacci length with respect to a generating pair $(x, y)$ is the fundamental period (if exist) of the truncated periodic sequence $z_0=x, z_1=y, z_k=z_{k-2}z_{k-1}, (k\geq 2)$ of the elements of $S$. We name this invariant as the Fibonacci invariant of $S$. Our used notation is the same as of the celebrated work of D.L. Johnson in $2005$ on infinite groups. In this paper we examine two classes of semigroups for existence of this invariant. The considered semigroups are the finite semigroup $S=\langle a, b\mid a^{p^\alpha}=a, b^{q^\beta}=b,ab=a\rangle$ of order $p^{\alpha}q^{\beta}-1$ and the infinite semigroup $T=\langle a, b\mid a^{p^\alpha}=a, ab=a\rangle$, for all integers $\alpha, \beta \geq 2$ and all distinct primes $p$ and $q$. We prove the existence of the Fibonacci invariants of these semigroups, for all parameters. As a numerical result we show that $\lambda (T)\leq \lambda (S)=p^{\alpha +1}$, if $p$ is even.

The Abelian Kernel of an Inverse Semigroup

The problem of computing the abelian kernel of a finite semigroup was first solved by Delgado describing an algorithm that decides whether a given element of a finite semigroup S belongs to the abelian kernel. Steinberg extended the result for any variety of abelian groups with decidable membership. In this paper, we used a completely different approach to complete these results by giving an exact description of the abelian kernel of an inverse semigroup. An abelian group that gives this abelian kernel was also constructed.

The structure of centralizers in the finite symmetric inverse semigroup

For an integer let be the symmetric inverse semigroup of partial injective transformations on an n-element set. For denote by and respectively, the first and second centralizer of λ in We determine the structure of and in terms of Green’s relations, including the partial order of -classes, and express as a direct product of cyclic groups with zero adjoined and a monogenic monoid. For each individual Green relation we determine such that in is inherited from and λ such that all Green relations in are inherited from We also provide a representation of the partial order of -classes in as a known lattice, and describe such that which gives a class of maximal commutative subsemigroups of

All 2-potent Elements in \(\mathit{Hyp}_G(2)\)

A generalized hypersubstitution of type \(\tau = (2)\) is a function which takes the binary operation symbol \(f\) to the term \(\sigma(f)\) which does not necessarily preserve the arity. Let \(Hyp_{G}(2)\) be the set of all these generalized hypersubstitutions of type \((2)\). The set \(Hyp_{G}(2)\) with a binary operation and the identity generalized hypersubstitution forms a monoid. The index and period of an element \(a\) of a finite semigroup are the smallest values of \(m\geq1\) and \(r\geq1\) such that \(a^{m+r}=a^m\). An element with the index \(m\) and period 1 is called an $m$-potent element. In this paper we determine all \(2\)-potent elements in \(Hyp_{G}(2)\).

Про матричнi зображення наднапiвгруп напiвгрупи, породженої двома взаємно анульовними iдемпотентами

Matrix representations of finite semigroups over fields are studied not so well as for finite groups. Representations of finite groups over fields are studied sufficiently well; in particular, the criterions of representation type are fully defined for an arbitrary field. If the characteristic $p$ of a field $K$ does not divide the order of a group (classical case), then the group has, up to equivalence, finite number of non-decomposable representations; such group is called a group of finite representation type. If the characteristic $p$ divides the order of a group (modular case), then the group has finite representation type only if its Sylow $p$-subgroup is cyclic. In this case for most finite groups the problems of describing their representations includes the problem on classification, up to similarity, of the pairs of matrices. Such groups are called wild, and groups that allow explicit descriptions of representations are called tame. The tame and wild groups in modular case are fully described by the first author together with Yu. A. Drozd. In the theory of representations of semigroups, the largest number of works are devoted to irreducible representations. Among the old results, there are only some results on semigroups of finite representation type, namely, for a finite quite simple semigroup (I. S. Ponizovsky) and some semigroups of all transformations of a finite set (I. S. Ponizovsky, C. Ringel). In cases, when the numbers of non-decomposable representations is infinite, the most famous are the results from the theory of representations of algebras that can be reformulated in terms of representations of semigroups: the description of representations of the algebra $ $ (I. M. Gelfand, V. A. Ponomarev and L. O. Nazarova, A. V. Roiter, V. V. Sergeichuk, V. M. Bondarenko) and the algebra $ $ (V. M. Bondarenko and C. Ringel). If we are not talking about individual semigroups, but about semigroup classes, then it should be noted works about on representations of the semigroups generated by idempotents with partial zero multiplication (V. M. Bondarenko, O. M. Tertychna), representations of the Rees semigroups (S. M. Dyachenko), semigroups generated by the potential elements (V. M. Bondarenko, O. V. Zubaruk) and representations of direct products of the symmetric second-order semigroup (V. M. Bondarenko, E. M. Kostyshyn). Such semigroups can have both a finite and infinite representation type. V. M. Bondarenko and Ja. V. Zatsikha described representation types of the third-order semigroups over a field, and indicate the canonical form of the matrix representations for any semigroup of finite representation type. This article is devoted to the study of similar problems for oversemigroups of commutative semigroups.

Finite monogenic semigroups and saturated varieties of semigroups

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.

Planar semilattices and nearlattices with eighty-three subnearlattices

Finite (upper) nearlattices are essentially the same mathematical entities as finite semilattices, finite commutative idempotent semigroups, finite join-enriched meet semilattices, and chopped lattices. We prove that if an nelement nearlattice has at least 83 · 2 n−8 subnearlattices, then it has a planar Hasse diagram. For n > 8, this result is sharp.

Self-injectivity of semigroup algebras

Abstract It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K 0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S] is a right (respectively, left) self-injective imply that S is finite? (Semigroup Algebras, Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K 0[S] being right (left) self-injective when K 0[S] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.