On the structure of IO n

We study the structure of the semigroup IO _n of all order-preserving partial bijections on an n-element set. For this semigroup we describe maximal subsemigroups, maximal inverse subsemigroups, automorphisms and maximal nilpotent subsemigroups. We also calculate the maximal cardinality for the nilpotent subsemigroups in IO _n which happens to be given by the n-th Catalan number.     

Structure of the semigroup <Emphasis Type="Italic">OT</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We study the structure of the semigroup OT _n , which is a unique (up to an isomorphism) R-section of the semigroup T _n . For this semigroup, we describe Green relations, determine regular and nilpotent elements, describe maximal nilpotent subsemigroups, and determine the unique irreducible system of generatrices and maximal subsemigroups.     

Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

We study mathematical models of the structure of nilpotent subsemigroups of the semigroup PTD(B _n ) of partial contracting transformations of a Boolean, the semigroup TD(B _n ) of full contracting transformations of a Boolean, and the inverse semigroup ISD(B _n ) of contracting transformations of a Boolean. We propose a convenient graphical representation of the semigroups considered. For each of these semigroups, the uniqueness of its maximal nilpotent subsemigroup is proved. For PTD(B _n ) and TD(B _n ) , the capacity of a maximal nilpotent subsemigroup is calculated. For ISD(B _n ), we construct estimates for the capacity of a maximal nilpotent subsemigroup and calculate this capacity for small n. For all indicated semigroups, we describe the structure of nilelements and maximal nilpotent subsemigroups of nilpotency degree k and determine the number of elements and subsemigroups for some special cases.     

A classification of the maximal idempotent-generated subsemigroups of finite singular semigroups

We describe the maximal idempotent-generated subsemigroups of the finite singular semigroup Sing _n on the finite set X _n ={1,2, \ldots,n} , and count the number of its maximal idempotent-generated subsemigroups. October 21, 1999     

Maximal Subsemigroups of Finite Transformation Semigroups K（n, r）

Let Tn be the full transformation semigroup on the n-element set Xn. For an arbitrary integer r such that 2 ≤ r ≤ n-1, we completely describe the maximal subsemigroups of the semigroup K(n, r) = {α∈Tn : |im α| ≤ r}. We also formulate the cardinal number of such subsemigroups which is an answer to Problem 46 of Tetrad in 1969, concerning the number of subsemigroups of Tn.     

A classification of maximal idempotent-generated subsemigroups of finite orientation-preserving singular partial transformation semigroups

In this paper we describe the maximal idempotent-generated subsemigroups of the finite orientation-preserving singular partial transformation semigroup SPOP _n and obtain their complete classification. We also obtain a classification of the maximal idempotent-generated subsemigroups of the finite order-preserving singular partial transformation semigroup PO^k_n\mathit{PO}^{k}_{n} with respect to ≤ _k .     

Maximal properties of some subsemigroups in finite order-preserving transformation semigroups

Let [n] = {1,2,…,n} be a finite set, ordered in the usual way. The order-preserving transformation semigroup O_n is the set of all order-preserving transformations of [n] (excluding the identity mapping) under composition. In this paper we first describe maximal idempotent-generated subsemigroups of O _n, and show that O_n has 2n - 2 such subsemi-groups. Secondly, we investigate maximal regular subsemigroups of O_n , and obtain the number of such subsemigroups as 2n - 3. Thirdly, we describe maximal idempotent-generated regular subsemigroups of O_n , and also obtain their classification and number.     

On nilpotent subsemigroups of the matrix semigroup over an antiring

In this paper, nilpotent subsemigroups in the matrix semigroup over a commutative antiring are discussed. Some basic properties and characterizations for the nilpotent subsemigroups are given, and some equivalent conditions for the matrix semigroup over a commutative antiring to have a maximal nilpotent subsemigroup are obtained. Also, the maximal nilpotent subsemigroups in the matrix semigroup are described.     

Structure of subsemigroups of factor-powers of finite symmetric groups

For the factor-powerFP(S _n ) of the symmetric groupS _n , we describe regular elements, maximal subgroups, isolated and fully isolated subsemigroups, and also maximal nilpotent subsemigroups whose zero elements coincide with the zero element of the semigroupFP(S _n ). Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 341–354, September, 1995. This research was partially supported by the Foundation for Fundamental Research of the State Committee for Science and Engineering of the Ukraine.     

Maximal Regular Subsemibands of Singn

We describe maximal regular subsemibands of the singular transformation semigroup Sing_n on the set X_n = {1, 2, . . . n} and obtain their complete classification We show that Sing_n has n(n + 1)/2 maximal regular subsemibands, and formulate the cardinal number of such subsemigroups.     

Maximal regular subsemibands of finite order-preserving transformation semigroups <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">r</Emphasis>)

Let O _n be the order-preserving transformation semigroup on X _n . For an arbitrary integer r such that 1≤r≤n−2, we completely describe the maximal regular subsemibands of the semigroup K(n,r)={α∈O _n :|im(α)|≤r}. We also formulate the cardinal number of such subsemigroups.     

Finite degree: algebras in general and semigroups in particular

An algebra A has finite degree if its term functions are determined by some finite set of finitary relations on A. We study this concept for finite algebras in general and for finite semigroups in particular. For example, we show that every finite nilpotent semigroup has finite degree (more generally, every finite algebra with bounded p _n -sequence), and every finite commutative semigroup has finite degree. We give an example of a five-element unary semigroup that has infinite degree. We also give examples to show that finite degree is not preserved in general under taking subalgebras, homomorphic images, direct products or subdirect factors.     

Maximal regular subsemigroups of certain semigroups of transformations

Let T _n and P _n be the full and partial transformation semigroups on a finite set of order n respectively. The properties of the subsemigroups of T _n and P _n have been widely studied. But the maximal regular subsemigroups of T _n and P _n seem to be unknown. In this note, we determine all the maximal regular subsemigroups of all ideals of T _n and P _n . April 7, 1999     

On existentially closed and generic nilpotent groups

Letn≧2 be an integer. We prove the following results that are known in casen=2: The upper and the lower central series of an existentially closed nilpotent group of classn coincide. A finitely generic nilpotent group of classn is periodic and the center of a finitely generic torsion-free nilpotent group of classn is isomorphic toQ ⁺, whereas infinitely generic nilpotent groups do not enjoy these properties. We determine the structure of the torsion subgroup of existentially closed nilpotent groups of class 2. Finally we give an algebraic proof that there exist 2^κ non-isomorphic existentially closed nilpotent groups of classn in cardinalityK ≧N ₀. Some results of this paper were contained in [6].     

The group-theoretic complexity of subsemigroups of boolean matrices

We find the group-theoretic complexity of many subsemigroups of the semigroup B_n of n × n Boolean matrices, including Hall matrices, reflexive matrices, fully indecomposable matrices, upper triangular matrices, row-rank-n matrices, and others.     

Maximal Normal Orthogroups in Rings Containing No Infinite Semilattices

Let R be a ring regarded as a multiplicative semigroup which contains no infinite subsemilattices. We investigate subsemigroups of R which are normal orthogroups, and present a construction from which all such maximal normal orthogroups can be obtained. In particular, we construct all maximal normal orthogroups of matrices over a field under matrix multiplication.

Communicated by D. Easdown.     

Bands of weakly r-Archimedean ordered semigroups

In this paper, some characterizations that an ordered semigroup S is a band of weakly r-archimedean ordered subsemigroups of S are given by some relations on S . We prove that an ordered semigroup S is a band of weakly r -archimedean ordered subsemigroups if and only if S is regular band of weakly r -archimedean ordered subsemigroups. Finally, we obtain that a negative ordered semigroup S is a band of weakly r-archimedean ordered subsemigroups of S if and only if S is a band of r-archimedean ordered subsemigroups of S . As an application the corresponding results on semigroups without order can be obtained by moderate modifications. August 27, 1999     

Perfect Semigroups and Aliens

Abstract. A topologized semigroup is called perfect if its multiplication is a perfect map (= a closed continuous mapping such that the inverse image of every point is compact). Thus a locally compact topological semigroup is perfect if and only if its multiplication is closed and each of its elements is compactly divided , that is, its divisors form a compact set. In the present paper we study compactly and non-compactly divided elements in the contexts of general locally compact semigroups, subsemigroups of groups, Lie semigroups and subsemigroups of Sl(2,R).     

On a Class of Hungarian Semigroups and the Factorization Theorem of Khinchin

Let G be a connected reductive Lie group and K be a maximal compact subgroup of G. We prove that the semigroup of all K-biinvariant probability measures on G is a strongly stable Hungarian semigroup. Combining with the result [see Rusza and Szekely⁽⁹⁾], we get that the factorization theorem of Khinchin holds for the aforementioned semigroup. We also prove that certain subsemigroups of K-biinvariant measures on G are Hungarian semigroups when G is a connected Lie group such that Ad G is almost algebraic and K is a maximal compact subgroup of G. We also prove a p-adic analogue of these results.