semillatice-ordered semigroups with apartness representation problem

Semilattice-ordered semigroups are important algebraic structures. they are ordered semigroups under partial order and ordered semigroups under anti-order. in this article we analyze constructive version of representation problem of semilattice-ordered semigroups with apartness in two theorems. difference between the classical case and the constructive case is in using the constructive logic instead the classical logic. besides, since algebraic structures here are supplied with diverse relations, we especially analyze how it affects the representation problem.