右自己移入的半群の絶対閉性について

Abstract

Hinkle [3] has showm that the direct product of column monomial matrix semigroups over groups is right self-injective. The author [12] has shown that the full transformation semigroup on a non-empty set (written on the left) is right self-injective and so every semigroup is embedded in a right self-injective regular semigroup. While absolutely closed semigroups has been first studied in Isbell [7]. In Howie and Isbell [51 and Scheiblich and Moore [8] it has been shown that inverse semigroups, totally division-ordered semigroups, right [left] simple semigroups, finite cyclic semigroups and full transformation semigroups are absolutely closed. In §1 we show that every right [left] self-injective semigrowp is absolutely closed. This gives alternative proofs that right [left] simple semigroups, finite cyclic semigroups and full transformation semigroups are absolutely closed. By using a result of [5] we show that the class of right [left] self-iujective [regular] semigrowps has the special amalgamation property. In §2 we show that a commutative separative semigroup is absolutely closed if and only if it is a semilattice of abelian groups. By using a characterization of self-injective inverse semigroups [9] we give a structure theorem for self-injective commutative separative semigroups.