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# Semigroup Forum (Online First™)

Articles from the last few issues of Semigroup Forum (Online First™)

## ✔ Generation of infinite factorizable inverse monoids

[CiTO]
Semigroup Forum (29 September 2011), pp. 1-17, doi:10.1007/s00233-011-9339-1

### Abstract

We investigate the generation of factorizable inverse monoids, paying special attention to the factorizable parts of the symmetric and dual symmetric inverse monoids. Key ideas covered include rank, relative rank, Sierpiński rank, and the semigroup Bergman property. The results for finite monoids are well-known or follow quickly from well-known facts, so most of the paper concerns the infinite case. ...

## ✔ The semigroup of combinatorial configurations

[CiTO]
Semigroup Forum (22 September 2011), pp. 1-6, doi:10.1007/s00233-011-9343-5

### Abstract

We elaborate on the existence and construction of the so-called combinatorial configurations. The main result is that for fixed degrees the existence of such configurations is given by a numerical semigroup. The proof is constructive giving a method to obtain combinatorial configurations with parameters large enough. ...

## ✔ On some medial semigroups with an associate subgroup

[CiTO]
Semigroup Forum (22 September 2011), pp. 1-35, doi:10.1007/s00233-010-9287-1

### Abstract

Let S be a semigroup and s , t ∈ S . We say that t is an associate of s if s = sts . If S has a maximal subgroup G such that every element s of S has a unique associate in G , say s ∗ , we say that G is an associate subgroup of S and consider the mapping s → s ∗ as a unary operation on S . In this way, semigroups with ...

## ✔ Special functions as subordinated semigroups on the real line

[CiTO]
Semigroup Forum (22 September 2011), pp. 1-17, doi:10.1007/s00233-011-9340-8

### Abstract

We give examples of convolution semigroups on the positive half-line and on the real line. Such semigroups are expressed in terms of special functions which arise in classical differential equations. ...

## ✔ Ordered semigroups which are both right commutative and right cancellative

[CiTO]
Semigroup Forum (22 September 2011), pp. 1-7, doi:10.1007/s00233-011-9346-2

### Abstract

In this paper we prove that each right commutative, right cancellative ordered semigroup ( S ,.,≤) can be embedded into a right cancellative ordered semigroup ( T ,○,⪯) such that ( T ,○) is left simple and right commutative. As a consequence, an ordered semigroup S which is both right commutative and right cancellative is embedded into an ordered semigroup T which is union of pairwise disjoint abelian groups, indexed by a left zero subsemigroup of T . ...

## ✔ Maximal regular subsemibands of finite order-preserving transformation semigroups <i>K</i>(<i>n</i>,<i>r</i>)

[CiTO]
Semigroup Forum (22 September 2011), pp. 1-19, doi:10.1007/s00233-011-9347-1

### Abstract

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 Tarski algebras are determined by their endomorphisms

[CiTO]
Semigroup Forum (21 September 2011), pp. 1-8, doi:10.1007/s00233-011-9345-3

## ✔ Green’s relations and regularity for semigroups of transformations that preserve reverse direction equivalence

[CiTO]
Semigroup Forum (21 September 2011), pp. 1-10, doi:10.1007/s00233-011-9344-4

### Abstract

Let T X denote the full transformation semigroup on a set X . For an equivalence E on X , let $$T_∃(X)={α∈ T_X:∀ x,y∈ X,(xα,yα)∈ E\Rightarrow(x,y)∈ E}.$$ Then T ∃ ( X ) is exactly the semigroup of mappings on the topological space X for which the collection of all E -classes is a basis. In this paper, we discuss regularity of elements and Green’s relations for T ∃ ( X ). ...

## ✔ Classification of some <i>τ</i>-congruence-free completely regular semigroups

[CiTO]
Semigroup Forum (13 September 2011), pp. 1-15, doi:10.1007/s00233-011-9342-6

### Abstract

Let τ be an equivalence relation on a semigroup. We introduce τ -congruence-free semigroups, extending the notion of congruence-free semigroups, and classify all completely regular semigroups which are τ -congruence-free, where τ is one of Green’s relations and respectively. Taking τ as as well as , this settles two open problems posed by M. Petrich and N.R. Reilly. ...

## ✔ Metrizability of Clifford topological semigroups

[CiTO]
Semigroup Forum (9 September 2011), pp. 1-7, doi:10.1007/s00233-011-9341-7

### Abstract

We prove that a countably compact Clifford topological semigroup S is metrizable if and only if the set E = e ∈ S : ee = e of idempotents of S is a metrizable G δ -set in S . ...

## ✔ Green’s relations and regularity for semigroups of transformations that preserve order and a double direction equivalence

[CiTO]
Semigroup Forum (8 September 2011), pp. 1-10, doi:10.1007/s00233-011-9332-8

### Abstract

Let T X be the full transformation semigroup on a set X , $$T_E^*(X)={α∈ T_X:∀ x,y∈ X, (x,y)∈ E\Leftrightarrow (xα,yα)∈ E}$$ be the subsemigroup of T X determined by an equivalence E on X . In this paper the set X under consideration is a totally ordered set with n points. The set of all order preserving transformations in $$O_E^*(X)={α∈ T_E^*(X): ∀ x,y∈ X, x≤ y\Rightarrow xα≤ yα}.$$ In this paper, we discuss Green’s relations for and prove that is a ...

## ✔ Recurrence Theorem for Semigroup Actions

[CiTO]
Semigroup Forum (8 September 2011), pp. 1-20, doi:10.1007/s00233-011-9334-6

### Abstract

In this article the notion of Poincaré recurrence for semigroup actions is introduced. It recovers the well-known concept of recurrence in dynamical systems. The Poincaré recurrence theorem is extended from the setting of flows on metric spaces to the setting of semigroup actions on metric spaces. The results are applied to control systems and semigroups acting on fiber bundles. ...

## ✔ Sensitivity and chaos of semigroup actions

[CiTO]
Semigroup Forum (8 September 2011), pp. 1-10, doi:10.1007/s00233-011-9335-5

### Abstract

We prove that if an action of a C -semigroup S on a Polish space is syndetic transitive, then the system is either minimal and equicontinuous, or sensitive. Additionally, we show that if an action of an abelian monoid S on a Polish space has a transitive point x and a periodic orbit O such that is perfect where H = s ∈ S : s | O is an identity map, then the system is chaotic. ...

## ✔ Plactic algebra of rank 3

[CiTO]
Semigroup Forum (7 September 2011), pp. 1-26, doi:10.1007/s00233-011-9337-3

### Abstract

The structure of the algebra K [ M ] of the plactic monoid M of rank 3 over a field K is studied. The minimal prime ideals of K [ M ] are described. There are only two such ideals and each of them is a principal ideal determined by a homogeneous congruence on M . Moreover, in case K is uncountable and algebraically closed, the left and right primitive spectrum and the corresponding irreducible representations of the algebra K [ ...

## ✔ Minimal presentations for monoids with the ascending chain condition on principal ideals

[CiTO]
Semigroup Forum (3 September 2011), pp. 1-6, doi:10.1007/s00233-011-9336-4

### Abstract

We show that the natural way to extend several key results concerning minimal presentations for finitely generated commutative cancellative reduced monoids, is to replace the finitely generated condition by the ascending chain condition on principal ideals. ...

## ✔ Addendum to: Quasi-ideal transversals of abundant semigroups and spined products

[CiTO]
Semigroup Forum (31 August 2011), pp. 1-3, doi:10.1007/s00233-011-9338-2

## ✔ Using filters to describe congruences and band congruences of semigroups

[CiTO]
Semigroup Forum, Vol. 83, No. 2. (1 October 2011), pp. 320-334, doi:10.1007/s00233-011-9330-x

### Abstract

It is well known that the smallest semilattice congruence can be described via filters. We generalise this result to the smallest left (right) normal band congruences and also to arbitrary semilattice (left normal band, right normal band) congruences, describing them all via filters. To achieve this, we introduce filters relative to arbitrary quasiorders on a semigroup (traditional filters are filters relative to the smallest negative operation-compatible quasiorder). We study congruences which can be described via filters. We show that the lattice ...

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

[CiTO]
Semigroup Forum (31 August 2011), pp. 1-12, doi:10.1007/s00233-011-9333-7

### Abstract

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 with respect to ≤ k . ...

## ✔ Hyperbolicity of solution semigroups for linear neutral differential equations

[CiTO]
Semigroup Forum (24 August 2011), pp. 1-13, doi:10.1007/s00233-011-9329-3

### Abstract

Consider the linear neutral functional differential equation of the form $$≤ft{\beginarrayl@\quad l\frac∂∂ tFu_t=BFu_t+Φ u_t, & t≥ 0,\\[3pt]u_0(s)=φ(s),& s∈ [-r,0],\endarray\right.$$ where the function u (⋅) takes values in a Banach space X . Under appropriate conditions on the difference operator F and the delay operator Φ we prove that the solution semigroup for this equation is hyperbolic provided that the differential operator B generates a hyperbolic semigroup on X . ...

## ✔ On Weierstrass semigroups of double coverings of genus three curves

[CiTO]
Semigroup Forum (24 August 2011), pp. 1-10, doi:10.1007/s00233-011-9331-9

### Abstract

Let C be a complete non-singular curve of genus 3 over an algebraically closed field of characteristic 0. We determine all possible Wierstrass semigroups of ramification points on double coverings of C whose covering curves have genus greater than 8. Moreover, we construct double coverings with ramification points whose Weierstrass semigroups are the possible ones. ...

## ✔ On the coset laws for skew lattices

[CiTO]
Semigroup Forum (18 August 2011), pp. 1-17, doi:10.1007/s00233-011-9325-7

### Abstract

Skew lattices are a noncommutative generalization of lattices. In the paper we study the varieties of symmetric, strongly symmetric and cancellative skew lattices, and characterize them in terms of certain laws regarding the coset structure of a skew lattice. As a consequence, combinatorial results connecting powers of -classes and indices are derived. ...

## ✔ The Frobenius problem for numerical semigroups with multiplicity four

[CiTO]
Semigroup Forum (3 August 2011), pp. 1-11, doi:10.1007/s00233-011-9328-4

### Abstract

In this paper, we study the gender, Frobenius number and pseudo-Frobenius number for numerical semigroups with multiplicity four, embedding dimension three and minimal generators pairwise relatively prime. ...

## ✔ A simple counterexample related to the Lie–Trotter product formula

[CiTO]
Semigroup Forum (3 August 2011), pp. 1-6, doi:10.1007/s00233-011-9326-6

### Abstract

In this note a very simple example is given which shows that if the sum of two semigroup generators is itself a generator, the generated semigroup in general can not be given by the Lie–Trotter product formula. ...

## ✔ Commutative semigroups with cancellation law: a representation theorem

[CiTO]
Semigroup Forum (27 July 2011), pp. 1-10, doi:10.1007/s00233-011-9327-5

### Abstract

Any commutative, cancellative semigroup S with 0 equipped with a uniformity can be embedded in a topological group . We introduce the notion of semigroup symmetry T which enables us to turn into an involutive group. In Theorem 2.8 we prove that if S is 2-torsion-free and T is 2-divisible then the decomposition of elements of into a sum of elements of the symmetric subgroup and the asymmetric subgroup is polar. In Theorem 3.7 we give conditions under which a topological ...

## ✔ Malcev products of unipotent monoids and varieties of bands

[CiTO]
Semigroup Forum (19 July 2011), pp. 1-29, doi:10.1007/s00233-011-9318-6

### Abstract

Let be the class of all unipotent monoids and the variety of all bands. We characterize the Malcev product where is a subvariety of low in its lattice of subvarieties, itself and the subquasivariety , where stands for semilattices and for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation , where ...

## ✔ Structure theorems for weakly <i>B</i>-abundant semigroups

[CiTO]
Semigroup Forum (12 July 2011), pp. 1-20, doi:10.1007/s00233-011-9323-9

### Abstract

The aim of this article is to provide structure theorems for weakly B-abundant semigroups satisfying the Congruence Condition (C), where B is a band. Such semigroups may be thought of as generalisations of orthodox semigroups. Our focus is on providing a description of a weakly B-abundant semigroup S with (C) as a spined product of a weakly B-abundant semigroup S B (depending only on B) and S/γ B , where γ B is the analogue of the ...

## ✔ A cohomology approach to the extension problem for commutative hypergroups

[CiTO]
Semigroup Forum (9 July 2011), pp. 1-24, doi:10.1007/s00233-011-9322-x

### Abstract

The purpose of this paper is to determine all commutative hypergroup extensions of a countable discrete commutative hypergroup by a locally compact Abelian group, in terms of second order cohomology of hypergroups, a notion which generalizes the cohomology of groups. ...

## ✔ Characterization of elements of polynomials in <i>βS</i>

[CiTO]
Semigroup Forum (1 July 2011), pp. 1-14, doi:10.1007/s00233-011-9321-y

### Abstract

Given the discrete space of natural numbers, we characterize the elements of polynomials evaluated on the points of βℕ. We establish these results by proving the characterization in a far more general setting. Let S be a discrete set which is a semigroup under two operations ⋅ and +. Let g(z 1,z 2,…,z k ) be any polynomial and p 1,p 2,…,p k be elements of βS. We provide a sufficient condition that a set A⊆S is a ...

## ✔ The regular part of a semigroup of transformations with restricted range

[CiTO]
Semigroup Forum (30 June 2011), pp. 1-13, doi:10.1007/s00233-011-9320-z

### Abstract

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed subset Y of X. It is known that $$F(X, Y)={α∈ T(X, Y): Xα⊆ Yα},$$ is the largest regular subsemigroup of T(X,Y) and determines Green’s relations on T(X,Y). In this paper, we show that F(X,Y)≅T(Z) if and only if X=Y and |Y|=|Z|; or |Y|=1=|Z|, and prove ...

## ✔ Optimal estimates for the semigroup generated by the classical Volterra operator on <i>L</i><sub><i>p</i></sub>-spaces

[CiTO]
Semigroup Forum (29 June 2011), pp. 1-8, doi:10.1007/s00233-011-9317-7

### Abstract

Optimal upper bounds are given for the norm of the semigroup (e −tV ) t≥0, where V is the classical Volterra operator acting on L p [0,1], 1≤p≤∞. In particular, for every p∈[1,∞] we prove that $$\mathopoverline\lim_t\to+∞\,≤ft(t^-|1/4-1/(2p)||e^-tV|_L_p\right)>0.$$ ...

## ✔ Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

[CiTO]
Semigroup Forum (24 June 2011), pp. 1-11, doi:10.1007/s00233-011-9319-5

### Abstract

We study algebraic and topological properties of the convolution semigroup of probability measures on a topological groups and show that a compact Clifford topological semigroup S embeds into the convolution semigroup P(G) over some topological group G if and only if S embeds into the semigroup of compact subsets of G if and only if S is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup S embeds into the functor-semigroup F(G) over ...

## ✔ Bundle structure of the Green classes in <img src="/fulltext-image.asp?format=htmlnonpaginated&src=Q781432328Q53839_html\233_2011_9314_Article_IEq1.gif" border="0" alt="$M_n(\mathbbK)$" />

[CiTO]
Semigroup Forum (21 June 2011), pp. 1-12, doi:10.1007/s00233-011-9314-x

### Abstract

It is shown using the semigroup structure of that the Green classes of the multiplicative semigroup of linear endomorphisms of an n-dimensional vector space V over (=ℝ, or ℂ) are the total spaces of certain fibre bundles having various Grassmann manifolds as the base spaces and the maps x↦R(x), x↦N(x) as the projection maps. In the case of a -class the fibre space is a certain Stiefel manifold and in the case of - and -classes the fibre ...

## ✔ A direct approach to the weighted admissibility of observation operators for bounded analytic semigroups

[CiTO]
Semigroup Forum (21 June 2011), pp. 1-13, doi:10.1007/s00233-011-9315-9

### Abstract

In this note we adopt the approach in Bonnit et al. (Czechoslov. Math. J. 60(2):527–539, 2010) to give a direct proof of some recent results in Haak and Le Merdy (Houst. J. Math., 2005) and Haak and Kunstmann (SIAM J. Control Optim. 45:2094–2118, 2007) which characterizes the L p -admissibility of type α depending on p of unbounded observation operators for bounded analytic semigroups. ...

## ✔ On semigroups admitting ring structure

[CiTO]
Semigroup Forum (17 June 2011), pp. 1-8, doi:10.1007/s00233-011-9316-8

### Abstract

A right-chain semigroup is a semigroup whose right ideals are totally ordered by set inclusion. The main result of this paper says that if S is a right-chain semigroup admitting a ring structure, then either S is a null semigroup with two elements or sS=S for some s∈S. Using this we give an elementary proof of Oman’s characterization of semigroups admitting a ring structure whose subsemigroups (containing zero) form a chain. We also apply this result, along with two other results ...

## ✔ Compact right multipliers on a Banach algebra related to locally compact semigroups

[CiTO]
Semigroup Forum (11 June 2011), pp. 1-9, doi:10.1007/s00233-011-9312-z

### Abstract

For a locally compact semigroup , let be the Banach space of all μ-measurable () functions vanishing at infinity, where denotes the algebra of all measures in the measure algebra of with continuous translations. Here, we study right compact multipliers on the Banach algebra equipped with an Arens product. ...

## ✔ Mehler semigroups, Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups

[CiTO]
Semigroup Forum (10 June 2011), pp. 1-27, doi:10.1007/s00233-011-9310-1

### Abstract

For finite dimensional vector spaces it is well-known that there exists a 1-1-correspondence between distributions of Ornstein-Uhlenbeck type processes (w.r.t. a fixed group of automorphisms) and (background driving) Lévy processes, hence between M- or skew convolution semigroups on the one hand and continuous convolution semigroups on the other. An analogous result could be proved for simply connected nilpotent Lie groups. Here we extend this correspondence to a class of commutative hypergroups. ...

## ✔ Finite degree: algebras in general and semigroups in particular

[CiTO]
Semigroup Forum (8 June 2011), pp. 1-22, doi:10.1007/s00233-011-9313-y

### Abstract

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 ...

## ✔ On regularity of sup-preserving maps: generalizing Zareckiĭ’s theorem

[CiTO]
Semigroup Forum (8 June 2011), pp. 1-7, doi:10.1007/s00233-011-9311-0

### Abstract

A sup-preserving map f between complete lattices L and M is regular if there exists a sup-preserving map g from M to L such that fgf=f. In the class of completely distributive lattices, this paper demonstrates a necessary and sufficient condition for f to be regular. When L=M is a power set, our theorem reduces to the well known Zareckiĭ’s theorem which characterizes regular elements in the semigroup of all binary relations on a set. Another application of our result is ...

## ✔ The product of quasi-ideal adequate transversals of an abundant semigroup

[CiTO]
Semigroup Forum (1 June 2011), pp. 1-9, doi:10.1007/s00233-011-9309-7

### Abstract

An inverse transversal of a regular semigroup S is an inverse subsemigroup that contains precisely one inverse of each element of S. This concept was first introduced by Blyth and McFadden and generalized to an adequate transversal in the abundant case by El-Qallali. In this paper we show that the product of any two quasi-ideal adequate transversals of an abundant semigroup S which satisfy the regularity condition is a quasi-ideal adequate transversal of S. Furthermore, all adequate transversals of S form a ...

## ✔ Clifford semigroups as functors and their cohomology

[CiTO]
Semigroup Forum (18 May 2011), pp. 1-14, doi:10.1007/s00233-011-9308-8

### Abstract

We prove that the category of Clifford semigroups and prehomomorphisms is isomorphic to a certain subcategory of the category of diagrams over groups. Under this isomorphism, Clifford semigroups are identified with certain functors. As an application of the isomorphism theorem, we show that the category with objects commutative inverse semigroups having the same semilattice of idempotents and with morphisms, the inverse semigroup homomorphisms that fix the semilattice, imbeds into a category of right modules over a certain ring. Also we ...

## ✔ On the separator of subsets of semigroups

[CiTO]
Semigroup Forum (30 April 2011), pp. 1-15, doi:10.1007/s00233-011-9306-x

### Abstract

By the separator of a subset A of a semigroup S we mean the set of all elements x of S which satisfy conditions xA⊆A, Ax⊆A, x(S−A)⊆(S−A), (S−A)x⊆(S−A). In this paper we deal with the separator of subsets of semigroups. In Sect. 2, we investigate the separator of subsets of special types of semigroups. We prove that, in the multiplicative semigroup S of all n×n matrices over a field and in the semigroup S of all transformations of ...

## ✔ Canonical semigroups

[CiTO]
Semigroup Forum (22 April 2011), pp. 1-10, doi:10.1007/s00233-011-9307-9

### Abstract

We define abstract canonical semigroups modeled after the canonical reductive monoids associated with the canonical compactification of a group of adjoint type. It then becomes possible for us to come up with semigroups having some of the algebraic properties of monoids of Lie type (without first starting with a group). ...

## ✔ A proof of Devadze’s theorem on generators of the semigroup of Boolean matrices

[CiTO]
Semigroup Forum (20 April 2011), pp. 1-8, doi:10.1007/s00233-011-9305-y

### Abstract

In 1968 Devadze described, without a proof, minimal sets of generators of the semigroup of n×n Boolean matrices. We provide a proof of Devadze’s theorem. ...

## ✔ Lattice isomorphisms of bisimple monogenic orthodox semigroups

[CiTO]
Semigroup Forum (15 April 2011), pp. 1-31, doi:10.1007/s00233-011-9304-z

### Abstract

Using the classification and description of the structure of bisimple monogenic orthodox semigroups obtained by the author (in Semigroup Forum 78, 310–325, 2009), we prove that every bisimple orthodox semigroup generated by a pair of mutually inverse elements of infinite order is strongly determined by the lattice of its subsemigroups in the class of all semigroups. This theorem substantially extends an earlier result of Shevrin (in Simon Stevin 67, 49–53, 1993) stating that the bicyclic semigroup is strongly lattice determined. ...

## ✔ Bernoulli measure on strings, and Thompson-Higman monoids

[CiTO]
Semigroup Forum (24 March 2011), pp. 1-32, doi:10.1007/s00233-011-9302-1

### Abstract

The Bernoulli measure on strings is used to define height functions for the dense - and -orders of the Thompson-Higman monoids M k,1. The measure can also be used to characterize the -relation of certain submonoids of M k,1. The computational complexity of computing the Bernoulli measure of certain sets, and in particular, of computing the - and -height of an element of M k,1 is investigated. ...

## ✔ Monoids that map onto the Thompson-Higman groups

[CiTO]
Semigroup Forum (23 March 2011), pp. 1-19, doi:10.1007/s00233-011-9303-0

### Abstract

A slight modification of the definition of the Thompson-Higman groups G k,1 and F k,1 leads to inverse monoids that map onto G k,1 (respectively F k,1), and that have interesting properties: they are finitely generated, and residually finite. These inverse monoids are closely related to the suffix expansion of G k,1 (respectively F k,1). ...

## ✔ <i>E</i>-unitary almost factorizable orthodox semigroups

[CiTO]
Semigroup Forum (22 March 2011), pp. 1-19, doi:10.1007/s00233-011-9299-5

### Abstract

It is established that an E-unitary almost factorizable orthodox semigroup need not be isomorphic to a semidirect product of a band by a group, and a necessary and sufficient condition is given for an E-unitary almost factorizable orthodox semigroup to be isomorphic to such a semidirect product. Moreover, the structure of every E-unitary almost factorizable orthodox semigroup is described by means of bands and groups. ...

## ✔ On the translational hull of a type B semigroup

[CiTO]
Semigroup Forum (17 March 2011), pp. 1-14, doi:10.1007/s00233-011-9301-2

### Abstract

In this paper, the translational hull of a type B semigroup is considered. We prove that the translational hull of a type B semigroup is itself a type B semigroup, and give some properties and characterizations of the translational hulls of such semigroups. Moreover, we consider the translational hulls of some special type B semigroups. These results strengthen the results of Fountain and Lawson (Semigroup Forum 32:79–86, 1985) on adequate semigroups. Finally, we give a new proof of a problem posted by Petrich ...

## ✔ Absolute flatness and amalgamation in pomonoids

[CiTO]
Semigroup Forum (16 March 2011), pp. 1-12, doi:10.1007/s00233-011-9300-3

### Abstract

Absolute flatness and amalgamation for partially ordered monoids (briefly pomonoids) were first considered in the mid 1980s by S.M. Fakhruddin in two research articles. Though the study of absolute flatness for pomonoids was revived by X. Shi, S. Bulman-Fleming and others after a dormancy period of almost two decades—resulting in the appearance of several research articles on the subject since 2005—amalgamation in pomonoids was never reconsidered until the recent past when S. Bulman-Fleming and the author produced two research articles on the subject. ...

## ✔ Optimization of classifiers for data mining based on combinatorial semigroups

[CiTO]
Semigroup Forum (24 February 2011), pp. 1-10, doi:10.1007/s00233-011-9298-6

### Abstract

The aim of the present article is to obtain a theoretical result essential for applications of combinatorial semigroups for the design of multiple classification systems in data mining. We consider a novel construction of multiple classification systems, or classifiers, combining several binary classifiers. The construction is based on combinatorial Rees matrix semigroups without any restrictions on the sandwich-matrix. Our main theorem gives a complete description of all optimal classifiers in this novel construction. ...

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