A subset $A$ of a semigroup $S$ is called a $chain$ ($antichain$) if $xy\in\{x,y\}$ ($xy\notin\{x,y\}$) for any (distinct) elements $x,y\in S$. A semigroup $S$ is called ($anti$)$chain$-$finite$ if $S$ contains no infinite (anti)chains. We prove that each antichain-finite semigroup $S$ is periodic and for every idempotent $e$ of $S$ the set $\sqrt[\infty]{e}=\{x\in S:\exists n\in\mathbb N\;\;(x^n=e)\}$ is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Also we present an example of an antichain-finite semilattice that is not a union of finitely many chains.
Given a semilattice $X$ we study the algebraic properties of the semigroup $\upsilon(X)$ of upfamilies on $X$. The semigroup $\upsilon(X)$ contains the Stone-Cech extension $β(X)$, the superextension $λ(X)$, and the space of filters $ϕ(X)$ on $X$ as closed subsemigroups. We prove that $\upsilon(X)$ is a semilattice iff $λ(X)$ is a semilattice iff $ϕ(X)$ is a semilattice iff the semilattice $X$ is finite and linearly ordered. We prove that the semigroup $β(X)$ is a band if and only if $X$ has no infinite antichains, and the semigroup $λ(X)$ is commutative if and only if $X$ is a bush with finit…
We review the recent approach to Markov chains using the Karnofksy-Rhodes and McCammond expansions in semigroup theory by the authors and illustrate them by two examples.
We investigate the maximum length of a chain of subsemigroups in various classes of semigroups, such as the full transformation semigroups, the general linear semigroups, and the semigroups of order-preserving transformations of finite chains. In some cases, we give lower bounds for the total number of subsemigroups of these semigroups. We give general results for finite completely regular and finite inverse semigroups. Wherever possible, we state our results in the greatest generality; in particular, we include infinite semigroups where the result is true for these. The length of a subgroup c…
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{P}_{n}$ be the semigroup of partial transformations on $[n]$. Let $\mathcal{CP}_{n}=\{α\in \mathcal{P}_{n}: (for ~all~x,y\in Dom~α)~|xα-yα|\leq|x-y|\}$ be the subsemigroup of partial contraction mappings on $[n]$. We have shown that the semigroup $\mathcal{CP}_{n}$ and some of its subsemigroups are nonregular left abundant semigroups for all $n$ but not right abundant for $n\geq 4$.
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{T}_{n}$ be the semigroup of full transformations on $[n]$. Let $\mathcal{CT}_{n}=\{α\in \mathcal{T}_{n}: (for ~all~x,y\in [n])~\left|xα-yα\right|\leq\left|x-y\right|\}$, then $\mathcal{CT}_{n}$ is a subsemigroup of $\mathcal{T}_{n}$. In this paper, we give a necessary and sufficient condition for an element to be regular and characterize all the Green's equivalences for the semigroup $\mathcal{CT}_{n}$. We further show that the semigroup $\mathcal{CT}_{n}$ is a left abundant semigroup.
We call a semigroup $S$ right noetherian if it satisfies the ascending chain condition on right ideals, and we say that $S$ satisfies ACCPR if it satisfies the ascending chain condition on principal right ideals. We investigate the behavior of these two conditions with respect to ideals and ideal extensions, with a particular focus on minimal and 0-minimal one-sided ideals. In particular, we show that the property of satisfying ACCPR is inherited by right and left ideals. On the other hand, we exhibit an example of a right noetherian semigroup with a minimal ideal that is not right noetherian.
Let $\mathcal{CP}_n$ be the semigroup of partial contraction mappings on $[n]=\{1,2,\ldots,n\}$ and let $\mathcal{OCP}_n$ and $\mathcal{ORCP}_n$ be its subsemigroups consisting of all order-preserving and of all order-preserving or order-reversing, partial contraction mappings, respectively. In this paper we obtain the rank of the two semigroups, $\mathcal{OCP}_n$ and $\mathcal{ORCP}_n$, respectively.
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{P}_{n}$ be the semigroup of partial transformations on $[n]$. Let $\mathcal{CP}_{n}=\{α\in \mathcal{P}_{n}: (for~all ~x,y\in Dom~α)~|xα-yα|\leq|x-y|\}$, then $\mathcal{CP}_{n}$ is a subsemigroup of $\mathcal{P}_{n}$. In this paper, we give a necessary and sufficient condition for an element in $\mathcal{P}_{n}$ to be regular and characterize all the Green's equivalences on the semigroup $\mathcal{CP}_{n}$.
Let $\mathcal{CT}_n$ be the semigroup of full contraction mappings on $[n]=\{1,2,\ldots,n\}$, and let $\mathcal{OCT}_n$ and $\mathcal{ODCT}_n$ be the subsemigroups consisting of all order-preserving full contraction and subsemigroup of order-decreasing and order-preserving full contraction mappings, respectively. In this paper, we show that the semigroup $\mathcal{ODCT}_n$ is left adequate. We further study the rank properties and as well obtain the rank of the semigroup, $\mathcal{ODCT}_n$. Moreover, we obtain a characterization of natural partial order for the semigroup $\mathcal{OCT}_n$ and…
Let ${\cal I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2, \ldots , n\}$ and let ${\cal DDP}_n$ and ${\cal ODDP}_n$ be its subsemigroups of order-decreasing partial isometries and of order-preserving and order-decreasing partial isometries of $X_n$, respectively. In this paper we investigate the cardinalities of some equivalences on ${\cal DDP}_n$ and ${\cal ODDP}_n$ which lead naturally to obtaining the order of the semigroups
Let ${\cal I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2,..., n\}$ and let ${\cal DP}_n$ and ${\cal ODP}_n$ be its subsemigroups of partial isometries and of order-preserving partial isometries of $X_n$, respectively. In this paper we investigate the cycle structure of a partial isometry and characterize the Green's relations on ${\cal DP}_n$ and ${\cal ODP}_n$. We show that ${\cal ODP}_n$ is a $0-E-unitary$ inverse semigroup. We also investigate the cardinalities of some equivalences on ${\cal DP}_n$ and ${\cal ODP}_n$ which lead naturally to obtaining the order of the semigroups.
Let ${\cal I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2,..., n\}$ and let ${\cal DDP}_n$ and ${\cal ODDP}_n$ be its subsemigroups of order-decreasing partial isometries and of order-preserving order-decreasing partial isometries of $X_n$, respectively. In this paper we investigate the cycle structure of order-decreasing partial isometry and characterize the Green's relations on ${\cal DDP}_n$ and ${\cal ODDP}_n$. We show that ${\cal ODDP}_n$ is a $0-E-unitary$ ample semigroup. We also investigate the cardinalities of some equivalences on ${\cal DDP}_n$ and ${\cal ODDP}_n$ which le…
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 $\mathscr{R}$…
Takahasi's theorem on chains of subgroups of bounded rank in a free group is generalized to several classes of semigroups. As an application, it is proved that the subsemigroups of periodic points are finitely generated and periodic orbits are bounded for arbitrary endomorphisms for various semigroups. Some of these results feature classes such as completely simple semigroups, Clifford semigroups or monoids defined by balanced one-relator presentations. In addition to the background on semigroups, proofs involve arguments over groups and finite automata.
The $\mathcal{R}$-height of a semigroup $S$ is the height of the poset of $\mathcal{R}$-classes of $S,$ i.e. the supremum of the lengths of chains of $\mathcal{R}$-classes. Given a semigroup $S$ with finite $\mathcal{R}$-height, we establish bounds on the $\mathcal{R}$-height of bi-ideals, one-sided ideals and two-sided ideals; in particular, these substructures inherit the property of having finite $\mathcal{R}$-height. We then investigate whether these bounds can be attained.
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 $\mathcal{R}$-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 $\mathcal{R}$- and $\mathcal{J}$-trivial semigroups.
A semigroup $S$ is called a weakly exponential semigroup if, for every couple $(a,b)\in S\times S$ and every positive integer $n$, there is a non-negative integer $m$ such that $(ab)^{n+m}=a^nb^n(ab)^m=(ab)^ma^nb^n$. A semigroup $S$ is called a $Δ$-semigroup if the lattice of all congruences of $S$ is a chain with respect to inclusion. The weakly exponential $Δ$-semigroups were described in [5]: A. Nagy, Weakly exponential $Δ$-semigroups, Semigroup Forum, 40(1990), 297-313. Although the existence of two types of them (T2R and T2L semigroups) is an open question, Theorem 3.11 of [5] gives neces…
The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the complex semigroup algebra is Frobenius, and so our results include applications to the study of Frobenius semigroup algebras. We explicitly factor the semigroup determinant for commutative semigroups and inverse semigroups. We recover the Wilf-Lindström factorization of the semigroup determinant of a meet semilattice and Wood's factorization for a finite commutat…
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.
Let $\mathcal C$ be a class of topological semigroups. A semigroup $X$ is called $absolutely$ $\mathcal C$-$closed$ if for any homomorphism $h:X\to Y$ to a topological semigroup $Y\in\mathcal C$, the image $h[X]$ is closed in $Y$. Let $\mathsf{T_{\!1}S}$, $\mathsf{T_{\!2}S}$, and $\mathsf{T_{\!z}S}$ be the classes of $T_1$, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup $X$ is absolutely $\mathsf{T_{\!z}S}$-closed if and only if $X$ is absolutely $\mathsf{T_{\!2}S}$-closed if and only if $X$ is chain-finite, bounded, group-…
In this paper we continue the investigations on the algebraic structure of a finite semigroup $S$ that is determined by its associated upper non-nilpotent graph $\mathcal{N}_{S}$. The vertices of this graph are the elements of $S$ and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of \B\ semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. …
The index of a subgroup of a group counts the number of cosets of that subgroup. A subgroup of finite index often shares structural properties with the group, and the existence of a subgroup of finite index with some particular property can therefore imply useful structural information for the overgroup. A developed theory of cosets in inverse semigroups exists, originally due to Schein: it is defined only for closed inverse subsemigroups, and the structural correspondences between an inverse semigroup and a closed inverse subsemigroup of finite index are weaker than in the group case. Neverth…
