Inner extensions of partial operations on a partial semigroup

Abstract

We analyse inner extensions of partial operations on a partial semigroup. The problem of extension of a partial operation internally to a full one with preservation of associativity is studied. The possibilities of continuing a partial operation on a partial semigroup of non-zero elements of a completely 0-simple semigroup by standard and non-standard methods are considered. A negative answer is obtained in relation to the question about whether any extension of a partial operation on a partial semigroup of non-zero elements is a completely simple semigroup, and whether any extension is standard. However, in certain cases the answers are positive. The article deduces the necessary and sufficient conditions of extendibility of a partial operation on a semigroup of residue modulo n, and also of a partial operation on a semigroup of non-zero elements of $(2\times 2)$-matrices over the field. The uniqueness of the extension of a partial operation on the semigroup of non-zero $(2\times 2)$-matrices over a field is shown.