An Introduction to Implicative Semigroups with Apartness

The setting of this research is Bishop’s constructive mathematics. Following ideas of Chan and Shum, exposed in their famous paper “Homomorphisms of implicative semigroups”, we discuss the structure of implicative semigroups on sets with tight apartness. Moreover, we use anti-orders instead of partial orders. We study concomitant issues induced by existence of apartness and anti-orders giving some specific characterizations of these semigroups. In addition, we introduce the notion of anti-filter in implicative semigroups and give some equivalent conditions that the inhabited real subset of an implicative semigroup is an ordered anti-filter.