Idempotent-Aided Factorizations of Regular Elements of a Semigroup

We establish the concept of idempotent-aided factorization (I.-A. factorization) of elements of a semigroup, as a semigroup-theoretical extension of full-rank factorization (F.-R. factorization) for matrices over a field. In the present paper, we introduce the concept of idempotent-aided factorization (I.-A. factorization) of a regular element of a semigroup, which can be understood as a semigroup-theoretical extension of full-rank factorization of matrices over a field. I.-A. factorization of a regular element d is defined by means of an idempotent e from its Green’s $\mathcal{D}$-class as decomposition into the product $d=uv$, so that the element u belongs to the Green’s $\mathcal{R}$-class of the element d and the Green’s $\mathcal{L}$-class of the idempotent e, while the element v belongs to the Green’s $\mathcal{L}$-class of the element d and the Green’s $\mathcal{R}$-class of the idempotent e. More precisely, it has been provenThe main result of the paper is a theorem which states that each regular element of a semigroup possesses an I.-A. factorization with respect to each idempotent from its Green’s $\mathcal{D}$-class. In addition, we prove that when one of the factors is given, then the other factor is uniquely determined. Several existence rules and characterizations of group inverses and $(b,c)$-inverses in a semigroup are provided based on proposed factorizations.I.-A. factorizations are then used to provide new existence conditions and characterizations of group inverses and $(b,c)$-inverses in a semigroup. In our further research, these factorizations will be applied to matrices with entries in a field, and efficient algorithms for realization of such factorizations will be provided.