Self-injective inverse semigroups (Group, Ring, Language and Related Areas in Computer Science)

Abstract

In the paper [l] and [2], it was shown that the injective hull of a semilattice is an 𝙎-distributive completion 𝘾(𝙎) in which 𝙎 is large. For an inverse semigroup 𝙎, B.Schein [3] gave a method of construction of 𝙎-distributive completion 𝘾(𝙎) of 𝙎. By a similar way of the construction of completions, B. Schein [5] described the injective hull of an inverse semigroup as the set 𝙁(𝙎) of all filters of 𝙎. B.Schein [4] showed that an inverse semigroup 𝙎 is self-injective if and only if 𝙎 is a 𝙎-distribute completion and 𝙀-reflexive. By making use of Schen's comletion, K.Shoji [6] showed that for any 𝙀-reflexive inverse semigroup 𝙎, there exists an inverse semigroup 𝙏 such that (1) 𝙎 is a subsemigroup of 𝙏, (2) 𝙏is the injective hull of 𝙎 as a right 𝙎-set and (3) 𝙏 is a self-injective semigroup. In this paper, we shall show that 𝙁(𝙎) is not always closed under set products.