On finite <Emphasis Type="Italic">X</Emphasis>-semilattices of unions

Using a characteristic family of sets, a characteristic mapping, and basis sources of an X-semilattice of unions D, we characterize the class Σ(X, m) consisting of all finite X-semilattices of unions that are isomorphic to a semilattice D given in advance. For a finite set X, the number of elements in the considered class is found. Commutative semigroups of idempotents are known to play a significant role in semigroup theory (see [25, 26]). Moreover, any commutative idempotent semigroup is isomorphic to some X-semilattice of unions (see [26]), whereas X-semilattices play an especially important role in studying many abstract properties of complete semigroups of binary relations (see [1–4, 7–24]). __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 27, Algebra and Geometry, 2005.     

Regular elements of semigroups B X ( D ) defined by generalized elementary B X ( D )-semilattices of unions

In the paper, the class of complete semigroups of binary relations is considered, each of whose elements is defined by a complete B _X (D)-semilattice of unions which belongs to the class of generalized elementary X-semilattices. Regular elements are described for each semigroup of this class. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 49, Algebra and Geometry, 2007.     

A Generalization of Symmetric Inverse Semigroups

The set of difunctional binary relations D_X plays a special role in representing inverse semigroups by binary relations. However, D_X is not an inverse semigroup either with the standard operation ∘, or with an alternative operation introduced in [6]. We introduce a new binary operation ⋄ on the set B_X of binary relations. We demonstrate that (D_X, ⋄) is an inverse semigroup, and the symmetric inverse semigroup (I_X, ∘) is a subsemigroup of (D_X,⋄).     

Idempotent elements of the semigroup corresponding to an X-semilattice of unions of the class Σ16(X, 6)

It is known that if we know all XI-subsemilattices of a given X-semilattice of unions, then we can determine all idempotent elements of the semigroup, and the structure of idempotent elements is characterized. In this work, we find idempotent elements of the semigroup corresponding to X-semilattices of unions of the class ??₁₆(X, 6). Moreover, we give formulas for the number of idempotent elements, where X is finite.     

Complete semigroups of binary relations defined by finite XI-semilattices of unions

We give conditions under which the X-semilattice of unions is an XI-semilattice, i.e., let D be a finite X-semilattice of unions. The complete semigroup of binary relation B _X (D) is defined by the XI-semilattice of unions if and only if V (D, ??) = D for some ?? ?? B _X (D).     

Irreducible generating sets of complete semigroups of unions B x (D) defined by semilattices of the class Σ2(X, 4)

In complete semigroups of unions B _x (D) defined by semilattices of class ??₂(X, 4), we selecte subsets of certain type on which equivalent binary relations are defined and by means of these relations, irreducible generating sets of the considered semigroups are described.     

Transformation semigroups preserving a binary relation

We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B _σ (X) in the case where σ(X) is a quasi-order.     

Fundamental semigroups having a band of idempotents

The construction by Hall of a fundamental orthodox semigroup W _B from a band B provides an important tool in the study of orthodox semigroups. We present here a semigroup S _B that plays the role of W _B for a class of semigroups having a band of idempotents B. Specifically, the semigroups we consider are weakly B-abundant and satisfy the congruence condition (C). Any orthodox semigroup S with E(S)=B lies in our class. On the other hand, if a semigroup S lies in our class, then S is Ehresmann if and only if B is a semilattice. The Hall semigroup W _B is a subsemigroup of S _B , as are the (weakly) idempotent connected semigroups V _B and U _B . We show how the structure of S _B can be used to extract information relating to arbitrary weakly B-abundant semigroups with (C). This work was carried out during a visit to Lisbon of the second author funded by the London Mathematical Society and while the first author was a member of project POCTI/0143/2003 of CAUL financed by FCT and FEDER.     

Dense inverse subsemigroups of a topological inverse semigroup

In this paper we study dense inverse subsemigroups of topological inverse semigroups. We construct a topological inverse semigroup from a semilattice. Finally, we give two examples of the closure of B _{( −∞, ∞ )}¹, a topological inverse semigroup obtained by starting with the real numbers as a semilattice with the operation a ∨ b=sup{a,b}. The author would like to thank to the referee for useful suggestions.     

The free idempotent generated locally inverse semigroup

In this paper we construct a model for the free idempotent generated locally inverse semigroup on a set X. The elements of this model are special vertex-labeled bipartite trees with a pair of distinguished vertices. To describe this model, we need first to introduce a variation of a model for the free pseudosemilattice on a set X presented in Auinger and Oliveira (On the variety of strict pseudosemilattices. Stud Sci Math Hungarica 50:207–241, 2013). A construction of a graph associated with a regular semigroup was presented in Brittenham et al. (Subgroups of free idempotent generated semigroups need not be free. J Algebra 321:3026–3042, 2009) in order to give a first example of a free regular idempotent generated semigroup on a biordered set E with non-free maximal subgroups. If G is the graph associated with the free pseudosemilattice on X, we shall see that the models we present for the free pseudosemilattice on X and for the free idempotent generated locally inverse semigroup on X are closely related with the graph G.     

Semilattices of bisimple inverse semigroups

The purpose of this paper is to give a structure for a semigroup which is a semilattice of bisimple inverse semigroups and satisfies certain conditions. For such a semigroup, we characterize the idempotent separating congruences.     

Domain theory and mirror properties in inverse semigroups

Inverse semigroups are a class of semigroups whose structure induces a compatible partial order. This partial order is examined so as to establish mirror properties between an inverse semigroup and the semilattice of its idempotent elements, such as continuity in the sense of domain theory.     

Binary relations as lattice isomorphisms

Summary In 1963, Zaretskiį established a one-to-one correspondence between the setB _X of binary relations on a set X and the set of triples of the form (W, ϕ, V) where W and V are certain lattices and ϕ: W→V is an isomorphism. We provide a multiplication for these triples making the Zaretskiį correspondence a semigroup isomorphism. In addition, we consider faithful representations ofB _X by pairs of partial transformations and also as the translational hull of its rectangular relations. Using these triples, we study idempotents, regular and completely regular elements and relationsH-equivalent to some relations with familiar properties such as reflexivity, transitivity, etc. Entrata in Redazione il 14 aprile 1998.     

Intuitionistic fuzzy semiprime ideals in ordered semigroups

We give characterizations of different classes of ordered semigroups by using intuitionistic fuzzy ideals. We prove that an ordered semigroup is regular if and only if every intuitionistic fuzzy left (respectively, right) ideal of S is idempotent. We also prove that an ordered semigroup S is intraregular if and only if every intuitionistic fuzzy two-sided ideal of S is idempotent. We give further characterizations of regular and intra-regular ordered semigroups in terms of intuitionistic fuzzy left (respectively, right) ideals. In conclusion of this paper we prove that an ordered semigroup S is left weakly regular if and only if every intuitionistic fuzzy left ideal of S is idempotent.     

Endomorphism Seminear-Rings of Brandt Semigroups

We consider the endomorphisms of a Brandt semigroup B _n and the semigroup of mappings E(B _n ) that they generate under pointwise composition. We describe all the elements of this semigroup, determine Green's relations, consider certain special types of mapping, which we can enumerate for each n, and give complete calculations for the size of E(B _n ) for small n.     

η-Simple semigroups without zero and η ∗-simple semigroups with a least non-zero idempotent

A semigroup S is called η-simple if S has no semilattice congruences except S×S. Tamura in (Semigroup Forum 24:77–82, 1982) studied η-simple semigroups with a unique idempotent. In the present paper we consider a more general situation, that is, we investigate η-simple semigroups (without zero) with a least idempotent. Moreover, we study η ^?-simple semigroups with zero which contain a least non-zero idempotent.