Products of idempotent endomorphisms of free acts of infinite rank

In 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (i.e., composite) of idempotent self-maps of that set. Using a wreath product construction introduced by V. Fleischer, the first-named author was recently able to describe products of idempotent endomorphisms of a freeS-act of finite rank whereS is any monoid. The purpose of the present paper is to extend this result to freeS-acts of infinite rank.Research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494     

A characterization of left cancellative monoids by flatness properties

Communicated by Boris M. Schein     

Semigroups that are factors of subdirectly irreducible semigroups by their monolith

Jeek and Kepka [4] proved that a universal algebra A with at least one at least binary operation is isomorphic to the factor of a subdirectly irreducible algebra B by its monolith if and only if the intersection of all of its (nonempty) ideals is nonempty, and that B may be chosen to be finite if A is finite. (By an ideal of A is meant a non-empty subset I of A such that f(a₁, . . . , a_n) I whenever f is an n-ary fundamental operation of A and a₁, . . . , a_n A are elements with a_i I for at least one index i.) In the present paper, we prove that if A is a semigroup, then B may be chosen also to be a semigroup, but that a finite semigroup need not be isomorphic to the factor of a finite subdirectly irreducible semigroup by its monolith.     

On V. Fleischer's characterization of absolutely flat monoids

In his paperCompletely flat monoids (Uh. Zap. Tartu Un-ta610 (1982), 38–52 (Russian)) V. Fleischer gives a characterization of the absolute flatness of a monoidS in terms of certain one-sided ideals and one-sided congruences ofS. In the present work an alternative, more direct proof of Fieischer's theorem is provided, and the result is used to show that the multiplicative monoid of any semisimple Artinian ring is absolutely flat.Research supported by Natural Sciences and Engineering Research Council grant A4494.Research supported by Natural Sciences and Engineering Research Council grant A9241.Presented by Boris M. Schein.     

Examples concerning absolute flatness and amalgamation in pomonoids

Flatness properties of acts over monoids have been studied for almost four decades and a substantial literature is now available on the subject. Analogous research dealing with partially ordered monoids acting on posets was begun in the 1980s in two papers by S.M. Fakhruddin, and, after a dormancy period of some 20 years, has recently been rekindled with the appearance of several research articles. In comparing flatness properties of S-acts and S-posets, it has been noted that the imposition of order results in severe restrictions as far as absolute flatness is concerned. For example, whereas every inverse monoid is absolutely flat (meaning all of its left and right acts are flat), even the three-element chain in its natural order, considered as a pomonoid, fails to have this property. It has long been understood that absolutely flat monoids, in particular, inverse monoids, are amalgamation bases in the class of all monoids. The purpose of the present article is to further investigate absolute flatness of pomonoids and to begin to study its connection with amalgamation in that context. T.E. Hall’s results, that amalgamation bases in the class of all monoids have the so-called representation extension property (REP), which in turn implies the right congruence extension property, are first adapted to the ordered context. A detailed study of the compatible orders (of which there are exactly 13) on the three-element chain semilattice U then reveals a wide range of possibilities: exactly four of these orders render U absolutely flat as a pomonoid, two more give it the right order-congruence extension property in every extension (RCEP) (but fail to make it an amalgamation base because of the failure of the ordered analogue of (REP)), and for the remaining seven, even (RCEP) fails.