A minimal non-extendable partial semigroup

This article discusses partial semigroups with a finite number of elements. Any partial semigroup can be extended to a full semigroup by adding elements to it, for example, a zero semigroup, in an external semigroup way. The author of the article is interested in the question of continuation of a partial semigroup without adding any elements to it in an internal semigroup way. The aim of this work is to find an internally non-extendable partial semigroup with a minimal number of elements. With increasing the number of elements in the set the number of partial groupoids on this set increases exponentially, and the number of partial semigroups among these partial groupoids is not known in advance. In order to find such partial semigroups it is necessary to use a computer or the Internet. In the Internet (GAP package) there are stored all the semigroups up to isomorphism and antiisomorphism on the set consisting of no more than 8 elements; that is why it will be enough to get partial semigroups out of semigroups with zero by deleting zero. The possibility of continuation of a partial semigroup in an internal semigroup way was checked out by a computer. As a result, it was revealed that all the partial semigroups on the set consisting of no more than 4 elements can be extended in an internal semigroup way to full ones. On the 5-element set, there is only one partial semigroup up to isomorphism and antiisomorphism, which can not be extended to a full semigroup.