Rees matrix semigroups over special structure groups with zero

Abstract

Let S be a semigroup with zero and let a S\O. Denote by V(a) the set of all inverses of a, that is, V(a) = (x ∈ S: axa=a. xax=x). Let n be a fixed positive integer. A semigroup S with zero is said to be homogeneous n regular if the cardinal number of the set V(a) of all inverses of a is n for every nonzero element a in S. Let T be a subset of S. We denote by E(T) the set of all idempotents of S in T.

The next theorem is a generalization of R. McFadden and Hans Schneider's theorem [1] .

Theorem 1. Let S be a 0-simple semigroup and let n be a fixed positive integer. Then the following are equivalent.

(i) S is a homogeneous n regular and completely 0-simple semigroup.

(ii) For every a≠0 in S there exist precisely n distinct nonzero elements (xᵢ)_{i [= symbol with an n on top]l} such that axᵢa=a for i=1, 2, ..., n and for all c, d in S cdc=c≠0 implies dcd=d.

(iii) For every a≠0 in S there exist precisely h distinct nonzero idempotents (eᵢ)_{i [= symbol with an h above]l} Eₐ and k distinct nonzero idempotents (fⱼ)_{j[= symbol with a k above]}= Fₐ such that eᵢa=a=afⱼ for i =1, 2, …, h, j = 1, 2, …, k hk=n, Eₐ contains every nonzero idempotent which is a left unit of a, Fₐ contains every nonzero idempotent which is a right unit of a and Eₐ ⋂ Fₐ contains at most one element.

(iv) For every a≠0 in S there exist precisely k nonzero principal right ideals (Rᵢ)_{i[= symbol with a k above]1} and h nonzero principal left ideals (Lⱼ)_{j[= symbol with h above]1} such that Rᵢ and Lⱼ contain h and k inverses of a, respectively, every inverse of a is contained in a suitable set Rᵢ ⋂ Lⱼ for i = 1, 2, .., k, j = 1, 2, .., h and Rᵢ ⋂ Lⱼ for i = 1, 2, .., k, j = 1, 2, .., h, and Rᵢ ⋂ Lⱼ contains at most one nonzero idempotent, where hk = n.

(v) Every nonzero principal right ideal R contains precisely h nonzero idempotents and every nonzero principal left ideal L contains precisely k nonzero idempotents such that hk=n, and R⋂L contains at most one nonzero idempotent.

(vi) S is completely 0-simple. For every 0-minimal right ideal R there exist precisely h 0-minimal left ideals (Li)_{i[= symbol with an h above]1} and for every 0-minimal left ideal L there exist precisely k 0-minimal right ideals (Rj)_{j[= symbol with a k above]1} such that LRⱼ=LiR=S, for every i=1,2,..,h, j=l,2,.. ,k, where hk=n.

(vii) S is completely 0-simple. Every 0-minimal right ideal R of S is the union of a right group with zero G°, a union of h disjoint groups except zero, and a zero subsemigroup Z uhich annihilates the right ideal R on the left and every 0-minimal left ideal L of S is the union of a left group with zero G’° a union of k disjoint groups except zero, and a zero subsemigroup Z' which annihilates the left ideal L on the right and hk=n.

(viii) S contains at least n nonzero distinct idempotents, and for every nonzero idempotent e there exists a set E of n distinct nonzero idempotents of S such that eE is a right zero subsemigroup of S containing precisely h nonzero idempotents, Ee is a left zero subsemigroup of S containing precisely k nonzero idempotents of S, e (E(S)\E) = (0) = (E(S)\E)e, and eE⋂Ee = (e), where hk=n.

S is said to be h-k type if every nonzero principal left ideal of S contains precisely k nonzero idempotents and every nonzero principal right ideal of S contains precisely h nonzero idempotents of S.

W. D. Munn defined the Brandt congruence [2]. A congruence ρ on a sernigroup S with zero is called a Brandt congruence if S/ρ is a Brandt semigroup.

Theorem 2. Let S be a 1-n type homogeneous n regular and complete:y 0-simple semigroup. Define a relation ρ on S in such a way that a ρb if and only if there exists a set (eᵢ) _{i[=symbol with an n above]1} of n distinct nonzero idempotents such that eᵢa=ebᵢ≠0, for every i=1, 2, . , n. Then ρ is an equivalence S\0. If we extend ρ on S by defining (0) to be ρ-class on S, then ρ is a proper Brandt congruence on S, then ρ ⊂ σ.

Let P=(pᵢⱼ) be any n x n matrix over a group with G°, and consider any n distinct points A₁, A₂, . , A_n in the plane, which we shall call vertices. For every nonzero entry pᵢⱼ≠0 of the matrix P, we connect the vertex Aᵢ to the vertex Aⱼ by means of a path [a bar over both AᵢAⱼ] which we shall call an edge (a loop if i = j) directed from Aᵢ to Aⱼ. In this way, with every n x n matrix P can be associated a finite directed graph G(P).

Let S=M°(G;In,In;P) be a Rees matrix semigroup. Then the graph G(P) is called the associated graph of the semigroup S, or simply it is the graph G(P) of S.

Theorem 3. A Rees matrix semigroup S=M°(G;In,In;P) is homogenous m² regular if the directed graph G(P) of the semigroup S is regular of degree m [3, p. 11].