Means, homomorphisms, and compactifications of weighted semitopological semigroups

We consider some almost periodic type function algebras on a weighted semitopological semigroup, and using the set of multiplicative means on each of these algebras, we define their corresponding weighted semigroup compactifications. This will constitute an effective tool for investigating the properties of the function algebras concerned. We also show that these compactifications do not retain all the nice properties of the ordinary semigroup compactifications unless we impose some restrictions on the weight functions.     

Invariant means,right ideals and the structure of semitopological semigroups

A number of functional analytic properties associated with the existence of a right invariant mean on a function spaceF over a semitopological semigroup S are given. We give topological and algebraic conditions on a right ideal I of S for which a classF(S) of functions on S has a right invariant mean if and only if the classF(I) has a right invariant mean.     

Problems on semigroups and control

Communicated by Karl H. Hofmann     

Compact semitopological inverse Clifford Semigroups

An inverse Clifford Semigroup is a semilattice of groups. Conditions are given for constructing a compact semitopological (separately continuous multiplication) inverse Clifford semigroup on a compact Hausdorff semilattice. The conditions are necessary and sufficient for decomposing a compact inverse Clifford semigroup containing a dense subgroup and locally compact maximal groups into its semilattice of groups. A catalogue of examples is given to demonstrate the construction while exhibiting various pathologies. This work was supported in part by the S.R.C.     

Problems and algorithms for affine semigroups

No abstract. December 21, 2000     

Some compactifications of a semitopological semigroup

Let S be a semitopological semgroup and let C_b (S) denote the B^*-algebra of all continuous bounded complex-valued functions on S. In this paper, we consider a left m-introverted and translation invariant B^*-subalgebra F of C_b(S) containing the constant functions. Our concern is with ΔF, the maximal ideal space of F up to an isomorphic homeomorphism, when it is made into a compact right topological semigroup containing a dense continous homomorphic image of S under the Gelfand topology and a suitably chosen binary operation. We establish a representation of the closed left ideals of ΔF and study its centre and ideal structure in some special cases.     

The Lawson number of a semitopological semilattice

Semigroup Forum - For a Hausdorff topologized semilattice X its Lawson number $$\bar{\Lambda }(X)$$ is the smallest cardinal $$\kappa $$ such that for any distinct points $$x,y\in X$$ there exists...     

Subgroups of products of metrizable semitopological groups

We give an internal characterization of subgroups of products of metrizable semitopological groups. This answers a question of M. Tkachenko.     

Metrizability of paratopological (semitopological) groups

We investigate some generalized metric space properties on paratopological (semitopological) groups and prove that a paratopological group that is quasi-metrizable by a left continuous, left-invariant quasi-metric is a topological group and give a negative answer to Ravsky?s question (Ravsky, 2001 [18, Question 3.1]). It is also shown that an uncountable paratopological group that is a closed image of a separable, locally compact metric space is a topological group. Finally, we discuss Hausdorff compactification of paratopological (semitopological) groups, give an affirmative answer to Lin and Shen?s question (Lin and Shen, 2011 [14, Question 6.9]) and improve an Arhangel?skii and Choban?s theorem. Some questions are posed.     

Quotient semigroups and extension semigroups

We discuss properties of quotient semigroup of abelian semigroup from the viewpoint of C ^*-algebra and apply them to a survey of extension semigroups. Certain interrelations among some equivalence relations of extensions are also considered.     

Invariant semigroups of orthodox semigroups

We consider, in a right inverse semigroupS with a multiplicative inverse transversalS ^o, the notion of anS ^o-invariant subsemigroup and use this to describe all the left amenable orders definable onS. The results obtained, together with their duals, are used to prove that ifS is an orthodox semigroup with a multiplicative inverse transversalS ^o, then every amenable order onS ^o can be extended to a unique amenable order onS. NATO Collaborative Research Grant 910765 is gratefully acknowledged. The second-named author also gratefully acknowledges support from the Calouste Gulbenkian Foundation, Lisbon.