The Binary Quasiorder on Semigroups (original) (raw)
Kragujevac J. Math. 27 (2005) 11–18. A NOTE OF A FAMILY OF QUASI-ANTIORDERS ON SEMIGROUP
2005
Abstract. Let (S,=, 6=, ·, s) be an ordered semigroup under an antiorder s. If S is a subdirect product of the ordered semigroup {Si: i ∈ I}, then there exists a family {σi: i ∈ I} of quasi-antiorders on S which separates the elements of S. Conversely, if {σi: i ∈ I} is a family of quasi-antiorders on S which separates the elements if S, then S is a subdirect product of the ordered semigroups {S/(σi ∪ (σi)−1) : i ∈ I}. This investigation is in constructive algebra. Throughout this paper, S = (S,=, 6=, ·) always denotes a semigroup with apartness in the sense of the books [1], [3], [8], [9] and papers [4], [5,], [6] and [7]. The apartness 6 = on S is a binary relation with the following properties ([1], [9]): For every elements x, y and z in S hold
On the Least (Ordered) Semilattice Congruence in Ordered Γ-Semigroups
Thai Journal of Mathematics, 2012
In this paper, we firstly characterize the relationship between the (ordered) filters,(ordered) s-prime ideals and (ordered) semilattice congruences in ordered Gamma\ Gamma Gamma-semigroups. Finally, we give some characterizations of semilattice congruences and ordered semilattice congruences on ordered Gamma\ Gamma Gamma-semigroups and prove that
In this paper, the notion of an ordered Γ-semigroup is introduced and some examples are given. Further the terms commutative ordered Γ-semigroup, quasi commutative ordered Γ-semigroup, normal ordered Γsemigroup, left pseudo commutative ordered Γsemigroup, right pseudo commutative ordered Γsemigroup are introduced. It is proved that (1) if S is a commutative ordered Γ-semigroup then S is a quasi commutative ordered Γ-semigroup, (2) if S is a quasi commutative ordered Γ-semigroup then S is a normal ordered Γ-semigroup, (3) if S is a commutative ordered Γ-semigroup, then S is both a left pseudo commutative and a right pseudo commutative ordered Γ-semigroup. Further the terms; left identity, right identity, identity, left zero, right zero, zero of an ordered Γ-semigroup are introduced. It is proved that if a is a left identity and b is a right identity of an ordered Γ-semigroup S, then a = b. It is also proved that any ordered Γsemigroup S has at most one identity. It is proved that if a is a left zero and b is a right zero of an ordered Γsemigroup S, then a = b and it is also proved that any ordered Γ-semigroup S has at most one zero element. The terms; ordered Γ-subsemigroup, ordered Γsubsemigroup generated by a subset, α-idempotent, Γ-idempotent, strongly idempotent, midunit, r-element, regular element, left regular element, right regular element, completely regular element, (α, β)-inverse of an element, semisimple element and intra regular element in an ordered Γ-semigroup are introduced. Further the terms idempotent ordered Γ-semigroup and generalized commutative ordered Γ-semigroup are introduced. It is proved that every α-idempotent element of an ordered Γ-semigroup is regular. It is also proved that, in an ordered Γ-semigroup, a is a regular element if and only if a has an ( , )-inverse. It is proved that, (1) if a is a completely regular element of an ordered Γ-semigroup S, then a is both left regular and right regular, (2) if "a" is a completely regular element of an ordered Γ-semigroup S, then a is regular and semisimple, (3) if "a" is a left regular element of an ordered Γ-semigroup S, then a is semisimple, (4) if "a" is a right regular element of an ordered Γ-semigroup S, then a is semisimple, (5) if "a" is a regular element of an ordered Γsemigroup S, then a is semisimple and (6) if "a" is a intra regular element of an ordered Γsemigroup S, then a is semisimple. The term separative ordered Γ-semigroup is introduced and it is proved that, in a separative ordered Γ-semigroup S, for any x, y, a, b ∈ S, the statements (i) xΓa ≤ xΓb if and only if aΓx ≤ bΓx, (ii) xΓ xΓa ≤ xΓxΓb implies xΓa ≤ xΓb,(iii) xΓyΓa ≤ xΓyΓb implies yΓxΓa ≤ yΓxΓb hold.
Semigroups with operation-compatible Green’s quasiorders
Semigroup Forum, 2016
We call a semigroup on which the Green's quasiorder ≤ J (≤ L , ≤ R) is operation-compatible, a ≤ J-compatible (≤ L-compatible, ≤ R-compatible) semigroup. We study the classes of ≤ J-compatible, ≤ L-compatible and ≤ R-compatible semigroups, using the smallest operation-compatible quasiorders containing Green's quasiorders as a tool. We prove a number of results, including the following. The class of ≤ L-compatible (≤ R-compatible) semigroups is closed under taking homomorphic images. A regular periodic semigroup is ≤ J-compatible if and only if it is a semilattice of simple semigroups. Every negatively orderable semigroup can be embedded into a negatively orderable ≤ J-compatible semigroup.