Connections between quasi-antiorder on a semigroup with apartness and a naturally defined quasi-antiorder relation on factor semigroup (according to congruence and anti-congruence) are presented. AMS Mathematics Subject Classification (2000): 03F55, 20M99

In this paper the concept of anti fuzzy ideals of a Γsemigroup has been introduced and some important characterizations have been obtained. Among other results a regular Γ-semigroup has been characterized in terms of anti fuzzy ideals. Finally we show that the set of all the anti fuzzy ideals of a Γ-semigroup forms a semilattice. 2010 AMS Classification: 20M99, 08A72, 20M12

In this paper various relationships between anti fuzzy ideals of a Γ-semigroup and that of its operator semigroups have been obtained. An inclusion preserving bijection between the set of all anti fuzzy ideals of a Γ-semigroup(not necessarily with unities) and that of its operator semigroups has been obtained. Mathematics Subject Classification: 20M99, 20N25

Generalized hypersubstitutions were introduced by S. Leeratanavalee and K. Denecke as a way of making precise the concepts of strong hyperidentity and M strong hyperidentity. A variety V is called M strongly solid if every identity of V is an M strong hyperidentity. We produce a monoid M for which the k-normalized variety Nv k (V ) of an N -strongly solid variety V with respect to a valuation v...

The semi-direct product of arbitrary two monoids and a presentation for this product have received considerable attention, see for instance [12], [14] and [15]. In [15], Wang defined a trivialiser set of the Squier complex associated with this presentation. In this paper, as a main result, we discuss necessary and sufficient conditions for the standard presentation of the semi-direct product of...

We prove that an hypersemigroup H is regular if and only, for any fuzzy subset f of H, we have f f ◦ 1 ◦ f and it is intra-regular if and only if, for any fuzzy subset f of H, we have f 1 ◦ f ◦ f ◦ 1. An hypersemigroup H is left (resp. right) quasi-regular if and only if, for any fuzzy subset f of S we have f 1 ◦ f ◦ 1 ◦ f (resp. f f ◦ 1 ◦ f ◦ 1) and it is semisimple if and only if, for any fuz...

The notion of absorption was developed a few years ago by Barto and Kozik and immediately found many applications, particularly in topics related to the constraint satisfaction problem. We investigate the behavior of absorption in semigroups and n-ary semigroups (that is, algebras with one n-ary associative operation). In the case of semigroups, we give a simple necessary and sufficient conditi...

By left magma-$e$-magma, I mean a set containingthe fixed element $e$, and equipped by two binary operations "$cdot$", $odot$ with the property $eodot (xcdot y)=eodot(xodot y)$, namelyleft $e$-join law. So, $(X,cdot,e,odot)$ is a left magma-$e$-magmaif and only if $(X,cdot)$, $(X,odot)$ are magmas (groupoids), $ein X$ and the left $e$-join law holds.Right (and two-sided) magma-$e$-magmas are de...

We observe simple links between preorders, semi-groups, rings and categories (and between equivalence relations, groups, fields and groupoids), which are type-definable in an arbitrary structure, and apply these observations to small structures. Recall that a structure is small if it has countably many pure n-types for each integer n. A ∅-type-definable group of finite arity in a small structur...

Let A' be a semisimple algebraic monoid with unit group G. Associated with E is its polyhedral root system (X, 0, C), where X = X(T) is the character group of the maximal torus T c G, $ c X(T) is the set of roots, and C = X(T) is the character monoid of T c E (Zariski closure). The correspondence £-»(A\ O, C) is a complete and discriminating invariant of the semisimple monoid £, extending the w...