Lower subsets of an ordered semigroup form in a natural way an ordered semigroup. This lower set operator gives an analogue of the power operator already studied in semigroup theory. We present a complete description of the lower set operator applied to varieties of ordered semigroups. We also obtain large families of fixed points for this operator applied to pseudovarieties of ordered semigrou...

In this paper, we present an algebraic approach to idempotent functional analysis, which is an abstract version of idempotent analysis in the sense of [1–3]. Elements of such an approach were used, for example, in [1, 4]. The basic concepts and results are expressed in purely algebraic terms. We consider idempotent versions of certain basic results of linear functional analysis , including the ...

In this paper, we introduced the concepts of roughness and fuzziness in ordered ternary semigroups. We proved that the lower and upper approximation of an ordered ternary subsemigroup (resp. left ideal, right ideal, lateral ideal, bi-ideal, interior ideal) in an ordered ternary semigroup is an ordered ternary subsemigroup (resp. left ideal, right ideal, lateral ideal, bi-ideal, interior ideal)....

We generalize recent results by G.A. Freiman, M. Herzog and coauthors on the structure theory of product-sets from the context of linearly (i.e., strictly and totally) ordered groups to linearly ordered semigroups. In particular, we find that if S is a finite subset of a linearly ordered semigroup generating a nonabelian subsemigroup, then |S2| ≥ 3 |S| − 2. On the road to this goal, we also pro...

In this paper, we apply the concept of fuzzy soft sets to ordered semigroup theory. The concepts of (∈γ ,∈γ ∨qδ)-fuzzy left (right) ideals, (∈γ ,∈γ ∨qδ)-fuzzy biideals and (∈γ ,∈γ ∨qδ)-fuzzy quasi-ideals are introduced and some related properties are obtained. Three kinds of lattice structures of the set of all (∈γ ,∈γ ∨qδ)-fuzzy soft left (right) ideals of an ordered semigroup are derived. The...