Actions of E-dense semigroups

In this work, we present a definition of an inverse semigroup, Pr(S), which is associated to an inverse semigroup S. Also, we show the existence of a kind of correspondence between partial representations of S and representations of Pr(S) on a Hilbert space. Some results of graded C*-algebras over a group are extended to pre-grading C*-algebras over inverse semigroups.

W.D. Munn proved that a finite-dimensional representation of an inverse semigroup is equivalent to a ∗-representation (by partial isometries) if and only if it is bounded. This paper gives a new analytic proof that every finite-dimensional representation of a compact inverse semigroup is equivalent to a ∗-representation. This will be the main result of this paper.

<p><strong>We study actions of inverse semigroups on arbitrary spaces, and relate properties of the action to topological properties of the associated groupoid of germs. Then, following the construction of Paterson in his 1999 book ``Inverse Semigroups, Groupoids and their Operator Algebras'', we investigate the intrinsic actions of inverse semigroups on their character spaces, and their groupoids of germs. We draw inspiration from a well-known result that characterizes sub-semigroups of the inverse semigroup of open bisections on an étale groupoid, which we call bisection wideness. Using Paterson's approach, we construct a number of ample groupoids associated to an inverse semigroup, these being the universal groupoid and the groupoid of ultragerms, and determine conditions under which an inverse semigroup S and a sub-semigroup W give rise to the same groupoid.</strong></p>

The inverse semigroup TE of isomorphisms of principal ideals of E onto principal ideals of E, where E is a semilattice, has been introduced and studied by Munn (1966, 1970). He showed that, for any inverse semigroup S with semilattice E, there is a representation of S by an inverse subsemigroup of TE. The Munn representation, however, is not always faithful. In this paper, the possibility is considered of enlarging the carrier set E of the Munn representation in order to obtain a faithful representation of S as an inverse subsemigroup of a structure resembling TE in many ways. A structure X is obtained by replacing each element of E by a set. Then X = ∪{Xe: e ∈ E}, where Xe, denotes some set, has a natural pre-order relation ≤ (where x ≤ y if and only if x ∈ Xe, y ∈ Xf and e ≦ f ) inherited from E such that if T = {(x, y)∈X × X;x ≤ y and y ≤ x} then X/T is isomorphic to E. Such a set X is referred to as a pre-semilattice with semilattice E. If Tx denotes the set of all isomorphisms of principal ideals of X onto principal ideals of X then Tx is an inverse semigroup. Basic properties of Tx are considered. It is shown that when X is locally uniform, that is, when |Xe| = |Xf|, for all e, f ∈ E, Tx may be described as a wreath product of a permutation group with TE.The set s itself is a presemilattice with semilattice E with respect to the pre-order ≤ defined by a ≤ b if and only if a−1a ≦ b−1b. It is then shown that the Vagner-Preston representation embeds S as a full inverse subsemigroup of Ts. As an application of these concepts the following result is established. Let R and S be inverse semigroups and let θ1(θ2) be an isomorphism of a semilattice E onto the semilattice of R(S). Then there exists a locally uniform presemilattice W and embeddings ϕ1, ϕ2 of R and S, respectively, as full inverse subsemigroups of Tw such that (1) θ1ϕ1 = θ2ϕ2 and (2) (eθ1ϕ1, eθ2ϕ2) ∈ if and only if Ee is isomorphic to Ef.

We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms which we termF-morphisms. These play the same role in the theory of idempotent-pure homomorphisms thatF-inverse monoids play in the theory ofE-unitary inverse semigroups.

We say that an action of a countable discrete inverse semigroup on a locally compact Hausdorff space is amenable if its groupoid of germs is amenable in the sense of Anantharaman-Delaroche and Renault. We then show that for a given inverse semigroup ${\mathcal{S}}$, the action of ${\mathcal{S}}$ on its spectrum is amenable if and only if every action of ${\mathcal{S}}$ is amenable.

Actions of inverse semigroups arising from partial actions of groups

We introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation α↦α+, where α+ is the identity map on the domain of α. We investigate the construction of ‘actions’ from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schröder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.

Partial actions and proper extensions of two-sided restriction semigroups

Let C C be a closed convex subset of a strictly convex Banach space X X and { T s : s ∈ S } \left \{ {{T_s}:s \in S} \right \} be a continuous representation of a semitopological semigroup S S as nonexpansive mappings of C C into itself. The main result establishes the fact that if for some x ∈ C x \in C the trajectory { T s x : s ∈ S } \left \{ {{T_s}x:s \in S} \right \} is relatively compact and A P ( S ) AP(S) has a left invariant mean then K = conv ⁡ { T s x : s ∈ S } ¯ K = \overline {\operatorname {conv} \{ {T_s}x:s \in S\} } contains a common fixed point for { T s } s ∈ S {\left \{ {{T_s}} \right \}_{s \in S}} .

In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gr\"othendieck group.

Using an algebraic point of view we present an introduction to the groupoid theory; that is, we give fundamental properties of groupoids as uniqueness of inverses and properties of the identities and study subgroupoids, wide subgroupoids, and normal subgroupoids. We also present the isomorphism theorems for groupoids and their applications and obtain the corresponding version of the Zassenhaus Lemma and the Jordan-Hölder theorem for groupoids. Finally, inspired by the Ehresmann-Schein-Nambooripad theorem we improve a result of R. Exel concerning a one-to-one correspondence between partial actions of groups and actions of inverse semigroups.

We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of “generalized weighted composition” operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is finitely generated then the underlying measure space is purely atomic, so that the semigroup is represented as “zero-unitary” matrices. The same is true if the semigroup contains a compact operator, in which case it is not even required that the semigroup be self-adjoint.

A theorem of Ostrowski implies that log(x), log(x +1), … are algebraically independent over ℂ(x). More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution y and particular transformations of y, such as derivatives of y with respect to parameters, shifts of the arguments, rescaling, etc. In this paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a ûnite collection of axioms that characterizes such actions to apply them to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are ûnite free products of semigroups of the form on Tannakian categories. This is the class of semigroups that appear in many applications.

In the field of transformational music theory, which emphasizes the possible transformations between musical objects, Klumpenhouwer networks (K-nets) constitute a useful framework with connections in both group theory and graph theory. Recent attempts at formalizing K-nets in their most general form have evidenced a deeper connection with category theory. These formalizations use diagrams in sets, i.e. functors where is often a small category, providing a general framework for the known group or monoid actions on musical objects. However, following the work of Douthett–Steinbach and Cohn, transformational music theory has also relied on the use of relations between sets of the musical elements. Thus, K-net formalizations should be extended further to take this aspect into account. The present article proposes a new framework called relational PK-nets, an extension of our previous work on poly-Klumpenhouwer networks (PK-nets), in which we consider diagrams in rather than . We illustrate the potential of relational PK-nets with selected examples, by analyzing pop music and revisiting the work of Douthett–Steinbach and Cohn.