A Construction for 𝒫-Regular Semigroups

Abstract

Let E(P) be a 𝒫-regular partial band and 𝒰 = {(q, p) ∈ P × P ∣ ⟨q⟩ ≃ _C⟨p⟩}. For any (q, p) ∈ 𝒰, the set of all C-isomorphisms from ⟨q⟩ onto ⟨p⟩ is denoted by C _q,p . The set C = ∪ {C _q,p  ∣ (q, p) ∈ 𝒰} forms a 𝒫-regular semigroup with P ⁺ = {θ _{p ∣ ⟨q⟩} ∣ p ∈ V _P (q)} as its C-set. Because it is a generalization of the Munn semigroup of a semilattice, C is referred to as the Munn semigroup of E(P). The properties of C(P ⁺) and the connection between C(P ⁺) and the Hall semigroup A(P*) of E(P) are discussed. Each 𝒫-regular semigroup is reconstructed with its greatest regular *-semigroup homomorphic image and the Munn semigroup of its 𝒫-regular partial band. This result generalizes results on orthodox semigroups. Noting also that C is a regular *-semigroup with P ^# = {I _⟨p⟩ ∣ p ∈ P} as the set of projections, a new structural description for strongly 𝒫-regular semigroups is obtained. This description only involves regular *-semigroups and avoids 𝒫-regular semigroups.

Keywords:

• Regular *-semigroup
• P-regular semigroup
• Strongly P-regular semigroup
• Munn semigroup
• Spined product

Acknowledgment

This work is supported by NSF of China (No. 10071068) and Youth Foundation of Hunan Education Committee (No. 99B14). The author would like to thank the referee for valuable comments on an earlier version of this paper.

Notes

^† Dedicated to my supervisor, Professor Yuqi Guo, on the occasion of his 60th birthday.