Extended Green’s relations on Mn(D), a characterization

In this paper, we consider the semiring [Formula: see text] of all [Formula: see text] matrices over a distributive lattice [Formula: see text] and extended Green’s relations [Formula: see text] and [Formula: see text] using [Formula: see text]-ideals. A (left, right) ideal [Formula: see text] of a semiring [Formula: see text] is called a (left, right) [Formula: see text]-ideal if [Formula: see text], where [Formula: see text]. We define [Formula: see text] and [Formula: see text] on a [Formula: see text]-regular semiring [Formula: see text], in which [Formula: see text] is a semilattice, as follows: [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the left [Formula: see text]-ideal generated by [Formula: see text] and [Formula: see text] is the right [Formula: see text]-ideal generated by [Formula: see text]. Here we characterize [Formula: see text] and [Formula: see text] in [Formula: see text] in terms of rows and columns of the matrices.

Solvability of a triple of binary relations with an application to green’s relations

In this paper, we introduce the notion of a solvable triple of binary relations on a set. This notion generalizes the notion of a regular relation and all other notions that are variants of the notion of the regularity, defined previously by many people. We also give some characterizations of the solvability of a triple of relations and use this to study Green’s relations on the monoid of binary relations on a set.

INJECTIVE LINEAR TRANSFORMATIONS WITH EQUAL GAP AND DEFECT

Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$ , we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $ , and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $ . We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$ . This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc.79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

Semigroups of an inductive composition of terms

The set of all [Formula: see text]-ary terms of type [Formula: see text] together with a binary operation derived from a superposition [Formula: see text] forms various forms of semigroups. One may generalize such binary operation by deriving it from an inductive composition of terms and call it an inductive product. However, this operation is not associative on the same base set but it becomes associative when all elements of subterms of a fixed term used in an inductive product except itself are excluded from the base set. Hence, a semigroup is formed. In this paper, we mainly focus on the algebraic structures of this semigroup such as idempotent elements, elements associating with each type of regularity condition, and Green’s relations. The formulae of complexity of inducted terms are also under investigation.

Fuzzy ∗–ideals and their applications in characterizing abundance and regularity of a semigroup

In this paper, the notion of a fuzzy *–ideal of a semigroup is introduced by exploiting generalized Green’s relations L * and R * , and some characterizations of fuzzy *–ideals on an arbitrary semigroup are obtained. Our main purpose is to establish the relationship between fuzzy *–ideals and abundance for an arbitrary semigroup. As an application of our results, we also give some new necessary and sufficient conditions for an arbitrary semigroup to be regular and inverse, respectively.

Green’s Relations on Regular Elements of Semigroup of Relational Hypersubstitutions for Algebraic Systems of Type ((m), (n))

Any relational hypersubstitution for algebraic systems of type (τ,τ′) = ((mi)i∈I,(nj)j∈J) is a mapping which maps any mi-ary operation symbol to an mi-ary term and maps any nj - ary relational symbol to an nj-ary relational term preserving arities, where I,J are indexed sets. Some algebraic properties of the monoid of all relational hypersubstitutions for algebraic systems of a special type, especially the characterization of its order and the set of all regular elements, were first studied by Phusanga and Koppitz[13] in 2018. In this paper, we study the Green’srelationsontheregularpartofthismonoidofaparticulartype(τ,τ′) = ((m),(n)), where m, n ≥ 2.

Congruences on bicyclic extensions of a linearly ordered group

In the paper we study inverse semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G) which are generated by partial monotone injective translations of a positive cone of a linearly ordered group G. We describe Green’s relations on the semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G), their bands and show that they are simple, and moreover, the semigroups B(G) and B^+(G) are bisimple. We show that for a commutative linearly ordered group G all non-trivial congruences on the semigroup B(G) (and B^+(G)) are group congruences if and only if the group G is archimedean. Also we describe the structure of group congruences on the semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G).