# Search Results

Consider the multiplicative semigroup M (S)of all probability distribution fu ctions on subsets S of R 2 .This structure corresponds to the coordi atewise maximum of S -valued independe t random vectors. We provide a wide class of possible territories S, where in spite of the lack of the u it element in M (S),there is a Khinchine-type decomposition.I case M (S)has a unit element,we characterize the subsets S with the property that there is a decompositio in M (S).

The class *CR* of completely regular semigroups considered as algebras with binary multiplication and unary operation of inversion forms a variety. Kernel, trace, local and core relations, denoted by **K**, **T**, **L** and **C**, respectively, are quite useful in studying the structure of the lattice *L*(*CR*) of subvarieties of *CR*. They are equivalence relations whose classes are intervals. Their ends are used for defining operators on *L*(*CR*).

Starting with a few band varieties, we repeatedly apply operators induced by upper ends of classes of these relations and characterize corresponding classes up to certain variety low in the lattice *L*(*CR*). We consider only varieties whose origin are “central” band varieties, that is those in the middle column of the lattice *L*(*B*) of band varieties. Several diagrams represent the (semi)lattices studied.

We characterize the domains of fractional powers of some operator matrices generating analytic semigroups. The results are applied to wave equations with delay and semilinear wave equations.

## Abstract

We develop the beginning of a theory of semigroups of linear operators on *p*-Frchet spaces, 0 < *p* < 1 (which are non-locally convex *F*-spaces), and give some applications.

## Abstract

We search for conditions on a countably compact (pseudocompact) topological semigroup under which: (i) each maximal subgroup
*H*(*e*) in *S* is a (closed) topological subgroup in *S*; (ii) the Clifford part *H*(*S*) (i.e. the union of all maximal subgroups) of the semigroup *S* is a closed subset in *S*; (iii) the inversion inv: *H*(*S*) → *H*(*S*) is continuous; and (iv) the projection π: *H*(*S*) → *E*(*S*), π: *x* ↦ *xx*
^{−1}, onto the subset of idempotents *E*(*S*) of *S*, is continuous.

## Abstract

We prove that all maximal subgroups of the free idempotent generated semigroup over a band *B* are free for all *B* belonging to a band variety **V** if and only if **V** consists either of left seminormal bands, or of right seminormal bands.

In the paper we give some remarks on the article of Janet Mills. In particular, the proof of Lemma 1.2 (in her work) is incorrect, and so the proof of Theorem 3.5 is not valid, too. Using different methods we show the mentioned theorem. Moreover, we find a new equivalent condition to the statements in Theorem 3.5. In particular, an explicit definition of a new class of orthodox semigroups is introduced.

In this paper we introduce the notions of essentially left amenable and approximately left amenable Lau algebras together with several characterizations of such algebras. In particular we investigate the relations between the left amenability, the essential left amenabil-ity and the approximate left amenability of certain quotient Lau algebras as well as operator projective tensor product of Lau algebras. The obtained results are applied to prove that the three notions coincide for a number of well-known semigroup algebras.

## Abstract

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