Lee and Kwon [12] defined an ordered semigroup S to be completely regular if a ∈ (a 2 Sa 2 ] for every a ∈ S. We characterize every completely regular ordered semigroup as a union of t-simple subsemigroups, and every Clifford ordered semigroup as a complete semilattice of t-simple subsemigroups. Green's Theorem for the completely regular ordered semigroups has been established. In an ordered semigroup S, we call an element e an ordered idempotent if it satisfies e ≤ e 2 . Different characterizations of the regular, completely regular and Clifford ordered semigroups are done by their ordered idempotents. Thus a foundation for the completely regular ordered semigroups and Clifford ordered semigroups has been developed. * correspondingauthor T. Saito studied systematically the influence of order on regular, idempotent, inverse, and completely regular semigroups [14] -[17], whereas Kehayopulu, Tsingelis, Cao and many others characterized regularity, complete regularity, etc. on ordered semigroups [1] -[3], [7]- [12]. Success attained by the school characterizing regularity on ordered semigroups are either in the semilattice and complete semilattice decompositions into different types of simple components, viz. left, t-, σ, λ-simple etc. or in its ideal theory.Complete regularity on ordered semigeoups was introduced by Lee and Kwon [12]. Here we give their complete semilattice decomposition and express them as a union of t-simple ordered semigroups. This supports the observation of Cao [3] that the t-simple ordered semigroups plays the same role in the theory of ordered semigroups as groups in the theory of semigroups without order. Then it follows that the semigroups which are semilattices of t-simple ordered semigroups are the analogue of Clifford semigroups. Though it is not under the name Clifford ordered semigroups, but such ordered semigroups have been studied extensively by Cao [2] and Kehayopulu [10], specially complete semilattice decomposition of such semigroups. We generalize such ordered semigroups into left Clifford ordered semigroups.Another successful part of this paper is identification of the ordered idempotent elements in an ordered semigroup and exploration of their behavior in both completely regular and Clifford ordered semigroups. Also an extensive study has been done on the idempotent ordered semigroups. Complete semilattice decomposition of these semigroups automatically suggests the looks of rectangular idempotent semigroups and in this way we arrive to many other important classes of idempotent ordered semigroups.The presentation of the article is as follows. This section is followed by preliminaries. In Section 3, basic properties of the t−simple ordered semigroups which we call here group like ordered semigroups have been studied.Completely regular ordered semigroups have been characterized in Section 4. Section 5 is devoted to the the Clifford ordered semigroups and their generalizations. PreliminariesAn ordered semigroup is a partially ordered set (S, ≤), and at the same time a semigrou...

No abstract

A regular ordered semigroup S is called right inverse if every principal left ideal of S is generated by an R-unique positive element of it. We prove that a regular ordered semigroup is right inverse if and only if any two inverses of an element a ∈ S are R-related. Furthermore the class of right Clifford ordered semigroups have been characterized by the class of right inverse ordered semigroups.

A code is called (n, k, r, t) information symbol locally repairable code (IS-LRC) if each information coordinate can be achieved by at least t disjoint repair sets containing at most r other coordinates. This letter considers a class of (n, k, r, t) IS-LRCs, where each repair set contains exactly one parity coordinate. We explore the systematic code in terms of the standard parity check matrix. First, we propose some structural features of the parity check matrix by showing a connection with the membership matrix. After that, we place parity check matrix based proof of several bounds associated with the code. In addition, we provide two constructions of optimal parameters of (n, k, r, t) IS-LRCs with the help of two Cayley tables of a finite field. Finally, we present a generalized result on optimal q-ary (n, k, r, t) IS-LRCs related to MDS codes.

No abstract