4,719 research outputs found
The Structure of Generalized BI-algebras and Weakening Relation Algebras
Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under Gumm–Ursini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for relation algebras together with two more terms. We prove that representable weakening relation algebras form a variety of cyclic involutive GBI-algebras, denoted by RWkRA, containing the variety of representable relation algebras. We describe a double-division conucleus construction on residuated lattices and on (cyclic involutive) GBI-algebras and show that it generalizes Comer’s double coset construction for relation algebras. Also, we explore how the double-division conucleus construction interacts with other class operators and in particular with variety generation. We focus on the fact that it preserves a special discriminator term, thus yielding interesting discriminator varieties of GBI-algebras, including RWkRA. To illustrate the generality of the variety of weakening relation algebras, we prove that all distributive lattice-ordered pregroups and hence all lattice-ordered groups embed, as residuated lattices, into representable weakening relation algebras on chains. Moreover, every representable weakening relation algebra is embedded in the algebra of all residuated maps on a doubly-algebraic distributive lattice. We give a number of other instructive examples that show how the double-division conucleus illuminates the structure of distributive involutive residuated lattices and GBI-algebras
Thr variety of coset relation algebras
A coset relation algebra is one embeddable into some full coset relation
algebra, the latter is an algebra constructed from a system of groups, a
coordinated system of isomorphisms between quotients of these groups, and a
system of cosets that are used to "shift" the operation of relative
multiplication. We prove that the class of coset relation algebras is
equationally axiomatizable (that is to say, it is a variety), but no finite set
of equations suffices to axiomatize the class (that is to say, the class is not
finitely axiomatizable).Comment: This is the fifth member of a series of papers on measurable relation
algebras. Forthcoming in The Journal of Symbolic Logic. arXiv admin note:
text overlap with arXiv:1804.0027
Conformal Covariance Subalgebras
We give a direct Lie algebraic characterisation of conformal inclusions of
chiral current algebras associated with compact, reductive Lie algebras. We use
straightforward quantum field theoretic arguments and prove a long standing
conjecture of Schellekens and Warner on grounds of unitarity and positivity of
energy. We explore the structures found to characterise ``conformal covariance
subalgebras'' and ``coset current algebras''.Comment: 9 pages, no figures; typos and minor improvement
W-Algebras of Negative Rank
Recently it has been discovered that the W-algebras (orbifold of) WD_n can be
defined even for negative integers n by an analytic continuation of their
coupling constants. In this letter we shall argue that also the algebras
WA_{-n-1} can be defined and are finitely generated. In addition, we show that
a surprising connection exists between already known W-algebras, for example
between the CP(k)-models and the U(1)-cosets of the generalized
Polyakov-Bershadsky-algebras.Comment: 12 papes, Latex, preprint DFTT-40/9
Dualizability of automatic algebras
We make a start on one of George McNulty's Dozen Easy Problems: "Which finite
automatic algebras are dualizable?" We give some necessary and some sufficient
conditions for dualizability. For example, we prove that a finite automatic
algebra is dualizable if its letters act as an abelian group of permutations on
its states. To illustrate the potential difficulty of the general problem, we
exhibit an infinite ascending chain of finite automatic algebras that are alternately dualizable and
non-dualizable
On ideals of a skew lattice
Ideals are one of the main topics of interest to the study of the order
structure of an algebra. Due to their nice properties, ideals have an important
role both in lattice theory and semigroup theory. Two natural concepts of ideal
can be derived, respectively, from the two concepts of order that arise in the
context of skew lattices. The correspondence between the ideals of a skew
lattice, derived from the preorder, and the ideals of its respective lattice
image is clear. Though, skew ideals, derived from the partial order, seem to be
closer to the specific nature of skew lattices. In this paper we review ideals
in skew lattices and discuss the intersection of this with the study of the
coset structure of a skew lattice.Comment: 16 page
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