Inner extensions of partial operations on a partial semigroup

  • Petrikov Alexander O., masterpetr@mail.ru National Research University of Electronic Technology, 1 Shokin Square, Zelenograd 124498, Russia
Keywords: inner extension, partial semigroup, completely 0-simple semigroup, semigroup of residue modulo n

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.

References

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How to Cite
Petrikov, A. ( ) “Inner extensions of partial operations on a partial semigroup”, Mathematical notes of NEFU, 24(2), pp. 30-45. doi: https://doi.org/10.25587/SVFU.2017.2.9244.
Section
Mathematics