# On a Class of Subdirect Products of Left and Right Clifford Semigroups

TL;DR: In this paper, the spined products of left and right Clifford semigroups are investigated in terms of spined product sparsification with respect to Clifford semiigroups with the property eS⊆Se or Se ⊆eS for all idempotents.

Abstract: Regular semigroups S with the property eS ⊆ Se or Se ⊆ eS for all idempotents e ∈ S include all left and right Clifford semigroups. Characterizations of such semigroups are given and their structure investigated, in particular in terms of spined products of left and right Clifford semigroups with respect to Clifford semigroups.

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01 Jan 1996

TL;DR: In this paper, it is shown that the above condition defining a reg-ular semigroup S to be a left (right) C-semigroup may be replaced by the following condi-tion:

Abstract: Left (right) C-semigroups are the regular semigroups satisfying the following condition:eSSe (SeeS) for every e∈E (the set of idempotents of S). Refs. [1, 2] give the left(right) semi-spined product structure and the left (right) △-product structure of a left(right) C-semigroup, respectively. It is easy to see that the above condition defining a reg-ular semigroup S to be a left (right) C-semigroup may be replaced by the following condi-tion:

8 citations

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TL;DR: In this article, the authors construct several commuting pairs of factor congruences which decompose any regular semigroup into a pullback product and give precise structural descriptions of the components in these pullback products.

Abstract: By an “effective” subdirect decomposition we mean a subdirect decomposition for which an effective construction of the corresponding family of factor congruences is given. In this paper we construct several commuting pairs of factor congruences which decompose any regular semigroup into a pullback product. For regular and completely regular semigroups whose idempotents form subsemigroups belonging to certain varieties of bands, we give precise structural descriptions of the components in these pullback products.

4 citations

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TL;DR: In this article, the authors introduced the notions of left (right) completely regular seminearrings and Clifford seminearring, and characterized them as bi-semilattices of left and right completely simple semearrings.

Abstract: In an attempt to investigate the situation arising out of replacing additive regularity by additive complete regularity in our previous study on additively regular seminearrings, we introduce the notions of left (right) completely regular seminearrings and characterize left (right) completely regular seminearrings as bi-semilattices of left (resp., right) completely simple seminearrings. We also define left (right) Clifford seminearrings and show that they are precisely bi-semilattices of near-rings (resp., zero-symmetric near-rings).

3 citations

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TL;DR: In this article, left k-Clifford semirings were introduced as a generalization of k-clifford semiirings and a semiring is a left-k-semifield semiring if and only if it is a distributive lattice of left-kliffords.

Abstract: Here, being motivated from the works of Zhu, Guo and Shum on left Clifford semigroups, we have introduced left k-Clifford semirings as a generalization of k-Clifford semirings. A k-regular semiring S ∈ 𝕊𝕃+ is a left k-Clifford semiring (left k-semifield) if for all a ∈ S, (for all b ∈ S′, ). Several characteristics for this class of semirings are obtained. Moreover a semiring is left k-Clifford if and only if it is a distributive lattice of left k-semifields.

2 citations

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TL;DR: A regular orthogroup S with the property that De=Re or De=Le for any idempotent e∈S is called a WLR-regular orthogroups as mentioned in this paper.

Abstract: A regular orthogroup S with the property that De=Re or De=Le for any idempotent e∈S is called a WLR-regular orthogroup. In this paper, we give constructions of such semigroups in terms of spined products of left and right regular orthogroups with respect to Clifford semigroups. WLR-cryptogroups and its special cases are also investigated.

2 citations

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TL;DR: The structure of general left-right inverse semigroups has been investigated in this paper, where it is shown that the set of idempotents of an orthodox semigroup S satisfies xyxzx=xyzx if and only if S is isomorphic to a subdirect product of a left inverse semiigroup and a right inverse semigroup.

Abstract: An orthodox semigroup S is called a left [right] inverse semigroup if the set of idempotents of S satisfies the identity xyx=xy [xyx=yx] Bisimple left [right] inverse semigroups have been studied by Venkatesan [6] In this paper, we clarify the structure of general left [right] inverse semigroups Further, we also investigate the structure of orthodox semigroups whose idempotents satisfy the identity xyxzx=xyzx In particular, it is shown that the set of idempotents of an orthodox semigroup S satisfies xyxzx=xyzx if and only if S is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup

34 citations

### "On a Class of Subdirect Products of..." refers background in this paper

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TL;DR: In this article, left Clifford semigroups are defined and ξ-products for such semiggroups and their semilattice decompositions are studied, and some structure theorems and characteristics for this class of semigroup are obtained.

Abstract: As generalization of Clifford semigroups, left Clifford semigroups are defined and ξ-products for such semigroups and their semilattice decompositions are studied. In particular, considering how a semilattice decomposition becomes a strong semilattice decomposition and ξ-product becomes spined product, some structure theorems and characteristics for this class of semigroups are obtained.

10 citations

### "On a Class of Subdirect Products of..." refers methods in this paper

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01 Jan 1996

TL;DR: In this paper, it is shown that the above condition defining a reg-ular semigroup S to be a left (right) C-semigroup may be replaced by the following condi-tion:

Abstract: Left (right) C-semigroups are the regular semigroups satisfying the following condition:eSSe (SeeS) for every e∈E (the set of idempotents of S). Refs. [1, 2] give the left(right) semi-spined product structure and the left (right) △-product structure of a left(right) C-semigroup, respectively. It is easy to see that the above condition defining a reg-ular semigroup S to be a left (right) C-semigroup may be replaced by the following condi-tion:

8 citations

### "On a Class of Subdirect Products of..." refers background in this paper

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