Frucht theorem for inverse semigroups

Abstract

ABSTRACT. In the paper, the problem of representing a finite inverse semigroup by partial transformations of a graph is treated. The notions of weighted graph and its weighted partial isomorphisms are introduced. The main result is that any finite inverse semigroup is isomorphic to the semigroup of weighted partial isomorphisms of a weighted graph. This assertion is a natural generalization of the Prucht theorem for groups. Since the sixties, other algebraic structures related to graphs were considered: the semigroups of endomorphisms of graphs, semigroups of partial endomorphisms, and so on (see the review [3]). The Kgnig problem makes sense for these structures as well. One of the recent results in this direction was obtained in [4] and consists of necessary and sufficient conditions for a semigroup of partial transformations of a set X to be a semigroup of partial endomorphisms of a graph with the set of vertices equal to X.Below by a graph we mean a finite nonoriented graph without loops. A partial automorphinm of a graph r is a one-to-one mapping (possibly not everywhere defined) Y(r) --, y(r) of the set V(r) of vertices of the graph F such that, for any vertices x and y from the domain domq0 of the mapping qo, the vertices q0(z) and ~p(y) are joined by and edge if and only if x and y are joined by an edge. The set PAut F of all partial automorphisms of the graph F forms a semigroup with respect to composition, and we can readily see that this semigroup is inverse. However, an arbitrary inverse semigroup is far from being isomorphic to PAut F for some graph F. One of the main reasons is that PAut F has too many ideals. In particular, PAut is never a group.A graph F is said to be weighted if a surjective mapping w: V(F) ~ P is given, where P is a lower semilattice; the value w(x) is called the weight of a vertez z E V(F). We say that a subset of vertices W C V(F) of a weighted graph F is a cone if W = {x E V(F) I w(z) <_ a} for some a E P. A partial automorphism qo E PAut F is said to be weighted if its domain dom ~ and its range ran qo are cones and preserves the ordering of weights of the vertices, i.e., for any a, b E dom qo, relation w(a) = w(b) implies w(~(a)) = w(qa(b)) and relation w(a) < w(b) implies w(qo(a)) < w (~(b)).Proposition. The set PAut~o F of all weighted automorphisms o/'a graph F is an inverse subsemigroup of the semigroup PAut P.Proof. It is clear that, for any q0 E PAut~ F, the element r that is inverse to qa belongs to PAut~ F as well, and the composition qor of elements from PA.ut~ F preserves the ordering of weights of the vertices. It remains to show that dom ~or and ran qa~b are cones. Let the cones ran q0 and domr be defined by elements a and ;3 of P, respectively. Since P is a lower semilattice, it follows from the definition of a cone that ran ~, n dom ~b = {x [ w(x) < ?}, This proves the proposition. D

Inverse monoids of partial graph automorphisms

“…This is in stark contrast to the well-known result of Frucht [6] who proved that every finite group is isomorphic to the automorphism group of a finite graph. A kind of 'extended' Frucht theorem for finite inverse monoids has been obtained in [22], where it is proved that every finite inverse monoid is isomorphic to the partial weighted automorphism monoid of a finite weighted graph. The construction takes a cleverly modified version of the Cayley color graph of an inverse monoid S and adds weights to the vertices in order to avoid the partial automorphisms not arising from the Wagner-Preston representation.…”

Inverse monoids of partial graph automorphisms

“…This is in stark contrast to the well-known result of Frucht [4] who proved that every finite group is isomorphic to the automorphism group of a finite graph. A kind of Frucht theorem has been obtained in [16], where it is proved that every finite inverse semigroup is isomorphic to the partial weighted automorphism monoid of a finite weighted graph, where a weighted graph is a graph whose vertices are assigned values from a lower semilattice, and a partial weighted automorphism is a partial graph automorphism, which preserves the weights of the vertices, and whose domain and range are required to be maximal sets of vertices with the property that their weights form a principal order ideal in the semilattice of weights. Given an inverse monoid S, the first step of the proof in constructing the appropriate weighted graph is to consider the Cayley color graph Γ of S, and to show that the Wagner-Preston representation of S consists of partial automorphisms of the edge-colored digraph Γ.…”