Boolean subsets of semigroups

Summary IfS→2 ^X is a surjective semigroup homomorphism, then the ultrafilters onX correspond biunivocally to the minimal prime ideals to the kernel. Some examples are given.

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Riassunto SeS→2 ^X è un omormorfismo surgettivo di semigruppi, allora gli ultrafiltri suX corrispondono biunivocamente agli ideali primi minimali del nucleo. Vengono dati degli esempi.

     

Invariant subsets of strongly continuous semigroups

Let {P(t): t0} be a strongly continuous semigroup on a Banach space X and let |\| be a continuous norm on X such that |P(t)x|exp(t)|x|, XX, t0. Let C be a |\|-closed convex subset of X and suppose that for every x in D(A) there exists a sequence (x_n : n ) in D(A) with the following properties: lim|x–x_n|=0, lim|Ax–Ax_n|=0 and every x_n has a best approximation in C (with respect to |\|) which belongs to D(A). Then P(t)CC for all t0 if and only if, for every v in CD(A), the vector Av belongs to the |\|-closure of [0, ) (C-V).     

Almost disjoint large subsets of semigroups

There are several notions of largeness in a semigroup S that originated in topological dynamics. Among these are thick, central, syndetic and piecewise syndetic. Of these, central sets are especially interesting because they are partition regular and are guaranteed to contain substantial combinatorial structure. It is known that in (N,+) any central set may be partitioned into infinitely many pairwise disjoint central sets. We extend this result to a large class of semigroups (including (N,+)) by showing that if S is a semigroup in this class which has cardinality κ then any central set can be partitioned into κ many pairwise disjoint central sets. We also show that for this same class of semigroups, if there exists a collection of μ almost disjoint subsets of any member S, then any central subset of S contains a collection of μ almost disjoint central sets. The same statement applies if “central” is replaced by “thick”; and in the case that the semigroup is left cancellative, “central” may be replaced by “piecewise syndetic”. The situation with respect to syndetic sets is much more restrictive. For example, there does not exist an uncountable collection of almost disjoint syndetic subsets of N. We investigate the extent to which syndetic sets can be split into disjoint syndetic sets.     

On extensions of the generalized quadratic functions from “large” subsets of semigroups

Let $$(G,+)$$ be a commutative semigroup, $$tau $$ be an endomorphism of G and involution, D be a nonempty subset of G, and $$(H,+)$$ be an abelian group, uniquely divisible by 2. Motivated by the extension problem of J. Aczél and the stability problem of S.M. Ulam, we show that if the set D is “sufficiently large”, then each function $$g{:} Drightarrow H$$ such that $$g(x+y)+g(x+tau (y))=2g(x)+2g(y)$$ for $$x,yin D$$ with $$x+y,x+tau (y)in D$$ can be extended to a unique solution $$f{:} Grightarrow H$$ of the functional equation $$f(x+y)+f(x+tau (y))=2f(x)+2f(y)$$.     

On U-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semi-groups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in HU on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup WU and a V-ample semigroup ...     

On Ehresmann semigroups

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right étale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.

     

On exclusive semigroups

The following problem was posed by the second author in ‘Semigroup Forum’, Vol. 1, No. 1, 1970, p. 91: Describe the structure of semigroups S such that for all a, b, c ε S, abc=ab, ac or bc.     

On implemented semigroups

We shall be concerned with implemented semigroups of the form U(t)X := T(t)XS(t) for X ∈ L(F,E) and t ≥ 0 , where (T(t)) _{t ≥ 0} and (S(t)) _{t ≥ 0} are C ₀ -semigroups on the Banach spaces E and F , respectively. The aim of this article is to clarify the structure of this semigroup and its ``generator' with the help of inter- and extrapolation techniques. Moreover, using positivity arguments, we present a short proof of the infinite dimensional Liapunov Stability Theorem on Hilbert spaces as an application of our results. May 15, 2000