Automatic presentations for semigroups

Special Issue: 2nd International Conference on Language and Automata Theory and Applications (LATA 2008)This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We give a complete list of FA-presentable one-relation semigroups and compare the classes of FA-presentable semigroups and automatic semigroups. (C) 2009 Elsevier Inc. All rights reserved.PostprintPeer reviewe

The Cayley isomorphism property for Cayley maps

The Cayley Isomorphism property for combinatorial objects was introduced by L. Babai in 1977. Since then it has been intensively studied for binary relational structures: graphs, digraphs, colored graphs etc. In this paper we study this prop- erty for oriented Cayley maps. A Cayley map is a Cayley graph provided by a cyclic rotation of its connection set. If the underlying graph is connected, then the map is an embedding of a Cayley graph into an oriented surface with the same cyclic rotation around every vertex. Two Cayley maps are called Cayley isomorphic if there exists a map isomorphism between them which is a group isomorphism too. We say that a finite group H is a CIM-group1 if any two Cayley maps over H are isomorphic if and only if they are Cayley isomorphic. The paper contains two main results regarding CIM-groups. The first one provides necessary conditons for being a CIM-group. It shows that a CIM-group should be one of the following ℤm × ℤr 2, ℤm × ℤ4, ℤm × ℤ8, ℤm × Q8, ℤm ⋊ ℤ2e, e = 1, 2, 3, where m is an odd square-free number and r a non-negative integer2. Our second main result shows that the groups ℤm × ℤr 2, ℤm × ℤ4, ℤm × Q8 contained in the above list are indeed CIM-groups. © 2018, Australian National University. All rights reserved

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Complex algebras of semigroups

The notion of a Boolean algebra with operators (BAO) was first defined by Jonsson and Tarski in 1951. Since that time, many varieties of BAOs, including modal algebras, closure algebras, monadic algebras, and of course relation algebras, have been studied. With the exception of relation algebras, these varieties each have one unary operator. This paper investigates a variety of BAOs with one binary operator. Begin with a semigroup S = (S,·). The complex algebra of S, denoted S+, is a Boolean algebra whose underlying set is the power set of S with set union, intersection and complementation as the Boolean algebra operations. The multiplication operation defined on the semigroup induces a normal, associative binary operation, \u27*\u27, on the complex algebra as follows: for all subsets A and B contained in S, A * B= a· b:a ϵ A, b ϵ B . Hence, the complex algebra of a semigroup is a BAO with one normal binary associative operator;Let S+ be the class of all complex algebras of semigroups and consider the variety generated by the class S+, denoted V(S+). The tools required to study this variety are developed, including the duality between BAOs and relational structures as it applies to V(S+). A closure operator is defined which is used to determine homomorphic images of members of V(S+). Theorems on the subdirectly irreducible and simples algebras in V(S+) are proved;Next, the structure of V(S+) is analyzed. The general problem of representing a BAO with one binary, normal, associative operator as a member of V(S+) is discussed. Several examples and theorems concerning representation are presented. It is shown that the quasivariety generated by S+ is strictly contained in V(S+). Lastly, the structure of the lattice of subvarieties of V(S+) is investigated. There are precisely two atoms in this lattice; each atom is generated by a two element algebra. Two infinite chains of varieties exist, with the smallest element in each chain a cover of exactly one atom. This leads to a discussion of certain splitting algebras and conjugate varieties. Finally, equations characterizing some important subvarieties are developed

Disjoint-union partial algebras

Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are defined. We provide a recursive first-order axiomatisation of the class of partial algebras isomorphic to a disjoint-union partial algebra of sets but prove that no finite axiomatisation exists. We do the same for other signatures including one or both of disjoint union and subset complement, another partial binary operation we define. Domain-disjoint union is a partial binary operation on partial functions, returning the union if the arguments have disjoint domains and undefined otherwise. For each signature including one or both of domain-disjoint union and subset complement and optionally including composition, we consider the class of partial algebras isomorphic to a collection of partial functions closed under the operations. Again the classes prove to be axiomatisable, but not finitely axiomatisable, in first-order logic. We define the notion of pairwise combinability. For each of the previously considered signatures, we examine the class isomorphic to a partial algebra of sets/partial functions under an isomorphism mapping arbitrary suprema of pairwise combinable sets to the corresponding disjoint unions. We prove that for each case the class is not closed under elementary equivalence. However, when intersection is added to any of the signatures considered, the isomorphism class of the partial algebras of sets is finitely axiomatisable and in each case we give such an axiomatisation.Comment: 30 page