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.     

Note on the orderability of idempotent semigroups

In [3] , we gave a condition for the orderability of finite idempotent semigroups. Recently, in [5], T. Saitô gives a condition for the orderability of idempotent semigroups in the general case. As Corollary ( 4–13 of [5] ), he obtains an idempotent semigroup S is orderable if and only if every finite subsemigroup of S is orderable. The purpose of this note is to give a direct proof of this result.     

Note on a certain class of orthodox semigroups

This is a continuation and also a supplement of the previous papers [5], [6] and [8] concerning orthodox semigroups¹). In [8], it has been shown that a quasi-inverse semigroup is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup. In this paper, we present a structure theorem for quasi-inverse semigroups and some relevant matters.     

Star-operations on semigroups

Let S be a grading monoid with quotient group q(S) , let F(S) be the set of fractional ideals of S . For A ∈ F(S) , define A _w = {x ∈ q(S) \mid J+x \subseteq A for some f.g. ideal J of S with J ^-1 =S} and A_ \overline w ={x ∈ q(S)\mid J+x \subseteq A for some ideal J of S with J ^-1 =S} . Then w and \overline w are star-operations on F(S) such that w ≤ \overline w . Using these star-operations, we give characterizations of Krull semigroups and pre-Krull semigroups. Also we show that for every maximal * -ideal P of S , if S _P is a valuation semigroup, then * -cancellation ideals are * -locally principal ideals, where * is a star-operation on S of finite character. Finally, we show that S is a pre-Krull semigroup (H-semigroup) if and only if the polynomial semigroup S[x] is a pre-Krull semigroup (H-semigroup). October 15, 1999     

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.     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

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.     

Coverages on inverse semigroups

Semigroup Forum - First we give a definition of a coverage on an inverse semigroup that is weaker than the one given by Lawson and Lenz and that generalizes the definition of a coverage on a...