It is well known that semigroups are basic structures in many applied branches like automata and formal languages, coding theory, finite state machines and others. Due to these possibilities of applications, semigroups and related structures are presently extensively investigated in soft theoretic (and in fuzzy theoretic) settings. The aim of this paper, is to introduce the concepts of double-framed soft ideals (briefly, DFS ideals) over an initial universe set U. The basic characterizations of this notion are studied. Different classes of ordered semigroups are characterized by using the notions of DFS l- and r-ideals over U. The notions of prime, and irreducible double-framed soft ideals (briefly, prime and irreducible DFS ideals) over U, are investigated and the related properties are highlighted. The class of those ordered semigroups for which each DFS ideal over U is prime are studied and the relationship between prime and irreducible DFS ideals are discussed.
Fully prime double-framed soft ordered semigroups