THE FIRST ISOMORPHISM THEOREM FOR (CO-ORDERED) $\Gamma$-SEMIGROUPS WITH APARTNESS

Abstract

The notion of $\Gamma$-semigroups has been introduced by Sen and Saha in 1986. This author introduced the concept of $\Gamma$-semigroups with apartness and analyzed
 their properties within the Bishop's constructive orientation. Many classical notions and processes of semigroups and $\Gamma$-semigroups have been extended to $\Gamma$-semigroups with apartness. Co-ordered $\Gamma$-semigroups with apartness have been studied by the author also. In this paper, as a continuation of previous research, the author investigates the specificity of two forms of the first isomorphism theorem for (co-ordered) $\Gamma$-semigroups with apartness which one of them has no a counterpart in the Classical case. In addition,
 specific techniques used in proofs within algebraic Bishop's constructive orientation are exposed.