Greatest regular images of tensor products of commutative semigroups
Tom Head, Nobuaki Kuroki · Kodai Mathematical Journal · 1975
Let A be a commutative semigroup which has either a greatest regular image or a greatest group image.Then for any commutative semigroup B, A®B has a greatest image of the same type and it is describable by standard constructions based on A and B. If a commutative semigroup A has a greatest group-with-zero image then A(g)B has such an image if and only if B is archimedean, in which case this image is again describable by standard constructions based on A and B, A handy elementary tool is the fact that the Grothendieck group of a commutative semigroup A may be regarded as the direct limit of the directed system of groups provided by Z(g)A where Z is the additive group of integers.By a type 2* of commutative semigroups we will mean a class of commutative semigroups that is closed under isomorphisms.We will deal with three types: the type of regular semigroups and two of its subtypes: groups and groups-with-zero.We say that a semigroup 5 has a greatest image of type £Γ if there is a homomorphism a of S onto a semigroup T in £Γ which is greatest in the sense that for every homomorphism β of S onto a semigroup U in H" we have β-ya for some homomorphism γ of T onto U.The purpose of the present article is to show that the possession of a greatest image by a commutative semigroup A may lead to the possession of a greatest image of the same type by tensor products of the form A® B. The study of tensor products of semigroups was initiated independently by three authors in [3], [4] and [6].Our work here may be regarded as a synthesis of [6] with the recent investigation of greatest regular images in [9].All semigroups considered will be commutative.Upper case letters will always denote commutative semigroups and Z will denote the additive group of integers.By a map we mean a semigroup homomorphism.1.The main results.For an arbitrary commutative semigroup A, Hewitt and Zuckerman [10] (or see [1, § 4.3]) described the construction of a regular