871 research outputs found
A method for finding new sets of axioms for classes of semigroups
We introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices
Identities in the Algebra of Partial Maps
We consider the identities of a variety of semigroup-related algebras modelling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable
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
Mathematical applications of inductive logic programming
Accepted versio
The rank of the semigroup of transformations stabilising a partition of a finite set
Let be a partition of a finite set . We say that a full
transformation preserves (or stabilizes) the partition
if for all there exists such that
. Let denote the semigroup of all full
transformations of that preserve the partition .
In 2005 Huisheng found an upper bound for the minimum size of the generating
sets of , when is a partition in which all of
its parts have the same size. In addition, Huisheng conjectured that his bound
was exact. In 2009 the first and last authors used representation theory to
completely solve Hisheng's conjecture.
The goal of this paper is to solve the much more complex problem of finding
the minimum size of the generating sets of , when
is an arbitrary partition. Again we use representation theory to
find the minimum number of elements needed to generate the wreath product of
finitely many symmetric groups, and then use this result to solve the problem.
The paper ends with a number of problems for experts in group and semigroup
theories
Causal structures and causal boundaries
We give an up-to-date perspective with a general overview of the theory of
causal properties, the derived causal structures, their classification and
applications, and the definition and construction of causal boundaries and of
causal symmetries, mostly for Lorentzian manifolds but also in more abstract
settings.Comment: Final version. To appear in Classical and Quantum Gravit
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