The Binary Quasiorder on Semigroups (original) (raw)

The binary quasiorder on EEE-central semigroups

2022

Given two elements x,yx,yx,y of a semigroup XXX we write xlesssimyx\lesssim yxlesssimy if for every homomorphism χ:Xto0,1χ:X\to\{0,1\}χ:Xto0,1 we have χ(x)leχ(y)χ(x)\leχ(y)χ(x)leχ(y). The quasiorder lesssim\lesssimlesssim is called the binarybinarybinary quasiorderquasiorderquasiorder on XXX. It induces the equivalence relation Updownarrow\UpdownarrowUpdownarrow that coincides with the least semilattice congruence on XXX. We discuss some known and new properties of the binary quasiorder and prove that for an EEE-central semigroup XXX the binary quasiorder induces the natural partial order on the semilattice E(X)E(X)E(X) of idempotents of XXX. This implies that for an EEE-central semigroup XXX, its subsemigroup UpdownarrowE(X){\Updownarrow}E(X)UpdownarrowE(X) admits canonical homomorphic retractions onto E(X)E(X)E(X) and onto the Clifford part H(X)=bigcupeinE(X)HeH(X)=\bigcup_{e\in E(X)}H_eH(X)=bigcupeinE(X)He of XXX.

A NOTE ON THE QUASI-ANTIORDER IN A SEMIGROUP

2007

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.

A note on quasi-antiorder in semigroup

Novi Sad J. Math, 2007

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.

The lattice of positive quasi-orders on a semigroup

Israel Journal of Mathematics, 1997

In the present paper we study some properties of positive quasi-orders on simigroups and using these results we describe all semilattice and chain homomorphic images of a semigroup.

A NOTE OF A FAMILY OF QUASI-ANTIORDERS ON SEMIGROUP

Kragujevac J. Math, 2005

Let (S, =, =, ·, s) be an ordered semigroup under an antiorder s. If S is a subdirect product of the ordered semigroup {S i : i ∈ I}, then there exists a family {σ i : i ∈ I} of quasi-antiorders on S which separates the elements of S. Conversely, if {σ i : 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/(σ i ∪ (σ i ) −1 ) : i ∈ I}. This investigation is in constructive algebra. Throughout this paper, S = (S, =, =, ·) 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 = on S is a binary relation with the following properties ([1], [9]): For every elements x, y and z in S hold

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