Congruence FD-maximal varieties of algebras

We study the class of finite lattices that are isomorphic to the congruence lattices of algebras from a given finitely generated congruence-distributive variety. If this class is as large as allowed by an obvious necessary condition, the variety is called congruence FD-maximal. The main results of this paper characterize some special congruence FD-maximal varieties

Bounding essential arities of term operations in finitely decidable varieties

Let $\mathbf{A}$ be a finite algebra generating a finitely decidable variety and having nontrivial strongly solvable radical $\tau$. We provide an improved bound on the number of variables in which a term can be sensitive to changes within $\tau$. We utilize a multi-sorted algebraic construction, amalgamating the methods developed by Valeriote and McKenzie for the investigation of strongly abelian locally finite decidable varieties with those of Idziak for locally finite congruence modular finitely decidable varieties

Distributive bilattices from the perspective of natural duality theory

This paper provides a fresh perspective on the representation of distributive bilattices and of related varieties. The techniques of naturalduality are employed to give, economically and in a uniform way, categories ofstructures dually equivalent to these varieties.We relate our dualities to the product representations for bilattices and to pre-existing dual representations by a simple translation process which is an instance of a more general mechanism for connecting dualities based on Priestley duality to natural dualities. Our approach gives us access to descriptions of algebraic/categorical properties of bilattices and also reveals how `truth' and `knowledge' may be seen as dual notions

Representation theory and homological stability

We introduce the idea of *representation stability* (and several variations) for a sequence of representations V_n of groups G_n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical theorems of homological stability to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood--Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the (n+1)^(n-1) conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures. Beyond the discovery of new phenomena, the viewpoint of representation stability can be useful in solving problems outside the theory. In addition to the applications given in this paper, it is applied in [CEF] to counting problems in number theory and finite group theory. Representation stability is also used in [C] to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds.Comment: 91 pages. v2: minor revisions throughout. v3: final version, to appear in Advances in Mathematic

Varieties whose finitely generated members are free

We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine vector spaces over a division ring.Comment: 17 page

Deformation rings and Hecke algebras in the totally real case

One of the basic questions in number theory is to determine semi-simple l-adic representations of the absolute Galois group of a number field. In this paper, we discuss the question for two dimensional representations over a totally real number field.Comment: This is the major update to my deformation ring paper, which is already submitted to a journa

Dynamical properties of logical substitutions

This is an expository paper on the dynamical properties of substitutions in propositional many-valued logics. We identify substitutions with endomorphisms of free algebras, and we study their actions on the dual spectral spaces.Comment: 23 pages, 5 figure

Restricted Priestley dualities and discriminator varieties

Anyone who has ever worked with a variety~$\boldsymbol{\mathscr{A}}$ of algebras with a reduct in the variety of bounded distributive lattices will know a restricted Priestley duality when they meet one---but until now there has been no abstract definition. Here we provide one. After deriving some basic properties of a restricted Priestley dual category $\boldsymbol{\mathscr{X}}$ of such a variety, we give a characterisation, in terms of $\boldsymbol{\mathscr{X}}$, of finitely generated discriminator subvarieties of~$\boldsymbol{\mathscr{A}}$. As a first application of our characterisation, we give a new proof of Sankappanavar's characterisation of finitely generated discriminator varieties of distributive double p-algebras. A substantial portion of the paper is devoted to the application of our results to Cornish algebras. A Cornish algebra is a bounded distributive lattice equipped with a family of unary operations each of which is either an endomorphism or a dual endomorphism of the bounded lattice. They are a natural generalisation of Ockham algebras, which have been extensively studied. We give an external necessary-and-sufficient condition and an easily applied, completely internal, sufficient condition for a finite set of finite Cornish algebras to share a common ternary discriminator term and so generate a discriminator variety. Our results give a characterisation of discriminator varieties of Ockham algebras as a special case, thereby yielding Davey, Nguyen and Pitkethly's characterisation of quasi-primal Ockham algebras

Clone Theory and Algebraic Logic

The concept of a clone is central to many branches of mathematics, such as universal algebra, algebraic logic, and lambda calculus. Abstractly a clone is a category with two objects such that one is a countably infinite power of the other. Left and right algebras over a clone are covariant and contravariant functors from the category to that of sets respectively. In this paper we show that first-order logic can be studied effectively using the notions of right and left algebras over a clone. It is easy to translate the classical treatment of logic into our setting and prove all the fundamental theorems of first-order theory algebraically

A duality for (n+1)-valued MV-algebras

MV-algebras were introduced by Chang to prove the completeness of the infinite-valued Lukasiewicz propositional calculus. In this paper we give a categorical equivalence between the varieties of (n+1)-valued MV-algebras and the classes of Boolean algebras endowed with a certain family of filters. Another similar categorical equivalence is given by A. Di Nola and A. Lettieri. Also, we point out the relations between this categorical equivalence and the duality established by R. Cignoli, which can be derived from results obtained by P. Niederkorn on natural dualities for varieties of MV-algebras.Comment: Reports on Mathematical Logic, 200