Search

An n-ary local polymorphism of a given monounary algebra A is a homomorphism from a finitely generated subalgebra of An to A. A is n-polymorphism-homogeneous if each n-ary local polymorphism can be extended to a global polymorphism. Then A is called polymorphism-homogeneous, if it is n-polymorphism-homogeneous for each positive integer n. In this paper we describe all polymorphism-homogeneous monounary algebras.

A theorem of single-sorted universal algebra asserts that every finite algebra can be represented as a product of a finite family of finite directly irreducible algebras. In this article, we show that the many-sorted counterpart of the above theorem is also true, but under the condition of requiring, in the definition of directly reducible many-sorted algebra, that the supports of the factors should be included in the support of the many-sorted algebra. Moreover, we show that the theorem of Birkhoff, according to which every single-sorted algebra is isomorphic to a subdirect product of subdirectly irreducible algebras, is also true in the field of many-sorted algebras.

The goal of this paper is to generalize a notion of quasi-characteristic inference rule (by using finite partial algebras instead of finite subdirectly irreducible algebras) in the following way: with every finite partial algebra we associate a (multiple-conclusion) rule, and study the properties of these rules. We prove that any equivalential logic can be axiomatized by such rules. We further discuss the correlations between characteristic rules of the finite partial algebras and canonical rules. Then, with every algebra we associate a set of characteristic rules that correspond to each finite partial subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular quasi-characteristic rules.

The Zariski closure of an arbitrary representable (not necessarily associative) algebra is studied in the general context of universal algebra, with an application being that the codimension sequence is exponentially bounded.

In this work we review the class T of ternary algebras introduced by J. A. Brzozowski and C. J. Serger in [1]. We determine properties of the congruence lattice of a ternary algebra A. The most important result refers to the construction of the free ternary algebra on a poset. In particular, we describe the poset of the join irreducible elements of the free ternary algebra with two free generators.

Some examples are given of applications of an in nite logical language in universal algebra, in the algebraic geometry of universal algebras, in the theory of implicit operations on algebras, in the Galois theory between automorphisms of universal algebras and its subalgebras of xed points, in the theory of Hamiltonian closure of subalgebras and other areas.

We present a complete description of subdirectly irreducible state BL-algebras as well as of subdirectly irreducible state-morphism BL-algebras. In addition, we present a general theory of state-morphism algebras, that is, algebras of general type with state-morphism which is a fixed idempotent endomorphism. We define a diagonal state-morphism algebra and we show that every subdirectly irreducible state-morphism algebra can be embedded into a diagonal one. We describe generators of the varieties of state-morphism algebras, in particular generators of state-morphism BL-algebras, state-morphism MTL-algebras, state-morphism non-associative BL-algebras, and state-morphism pseudo MV-algebras.

We present a characterisation of subalgebra primal algebras that are irreducible w.r.t. a localisation theory for finite algebras that replaces the role of polynomial operations and congruences in Tame Congruence Theory by term operations and arbitrary finitary compatible relations, respectively.

The equational complexity of Lyndon’s nonfinitely based 7-element algebra lies between n − 4 and 2n + 1. This result is based on a new algebraic proof that Lyndon’s algebra is not finitely based. We prove that Lyndon’s algebra is inherently nonfinitely based relative to a rather rich class of algebras. We also show that the variety generated by Lyndon’s algebra contains subdirectly irreducible algebras of all cardinalities except 0, 1, and 4.

We show that the full group C$^*$-algebra of the free product of two nontrivial countable amenable discrete groups, where at least one of them has more than two elements, is primitive. We also show that in many cases, this C$^*$-algebra is antiliminary and has an uncountable family of pairwise inequivalent, faithful irreducible representations., Comment: 18 pages. Preliminary version. Comments are wellcome.

We give a formula for the universal R R -matrix of the quantized universal enveloping algebra U q ( g ) . U_q(\mathfrak {g}). This is similar to a previous formula due to Kirillov-Reshetikhin and Levendorskii-Soibelman, except that where they use the action of the braid group element T w 0 T_{w_0} on each representation V V , we show that one can instead use a system of weight preserving endomorphisms. One advantage of our construction is that it is well defined for all symmetrizable Kac-Moody algebras. However, we have only established that the result is equal to the universal R R -matrix in finite type.

Let LB(−52,0) (resp. LF(−52,0)) be the simple vertex operator algebra associated with affine Lie algebra of type B4(1) (resp. F4(1)) with the lowest admissible half-integer level −52. We show that LB(−52,0) is a vertex subalgebra of LF(−52,0) with the same conformal vector, and that LF(−52,0) is isomorphic to the extension of LB(−52,0) by its only irreducible module other than itself. We also study the representation theory of LF(−52,0), and determine the decompositions of irreducible weak LF(−52,0)-modules from the category O into direct sums of irreducible weak LB(−52,0)-modules.

We consider here some notions and results about sets of generators of finite algebras motivated by the case of finite groups. We illustrate these notions by simple examples closed to lattices and semigroups. Next, we examine these notions in the case of groups with operators.

A Heyting effect algebra (HEA) is a lattice-ordered effect algebra that is at the same time a Heyting algebra and for which the Heyting center coincides with the effect-algebra center. Every HEA is both an MV-algebra and a Stone-Heyting algebra and is realized as the unit interval in its own universal group. We show that a necessary and sufficient condition that an effect algebra is an HEA is that its universal group has the central comparability and central Rickart properties.

We investigate ways of representing ordered sets as algebras and how the order relation is reflected in the algebraic properties of the variety (equational class) generated by these algebras. In particular we consider two different but related methods for constructing an algebra with one binary operation from an arbitrary ordered set with a top element. The two varieties generated by all these algebras are shown to be well-behaved in that they are locally finite, finitely based, and have an equationally definable order relation. We exhibit a bijection between the subdirectly irreducible algebras in each variety and the class of all ordered sets with top element. We determine the structure and cardinality of the free algebra on n-free generators and provide sharp bounds on the number of n-generated algebras in each variety. These enumeration results involve the number of quasi-orders on an n-element set.

The aim of this work is to study the universal Steenrod algebra Q p for odd primes p. An invariant-theoretic description of Q p is given and two families of algebras related to Q p are presented. They include the algebra Λ opp , the Steenrod algebra 𝒜 p and the Steenrod algebra for simplicial restricted Lie algebras A L as particular cases. The diagonal cohomology D*(Q p ) is computed.

We will study the Hall algebra ℋ(Δ) of hereditary orders. We will give an ideal I of such that is isomorphic to the universal enveloping algebra U(𝔤𝔩 n ) (Sec. 1.3). Then we will construct an ℋ(Δ)-module (m ∈ ℕ) by a Hall algebra approach, which restricts to the irreducible U(𝔰𝔩 n )-module with the highest weight mω1 (Sec. 1.4).

We generalize the notion of a monadic algebra to that of a pseudomonadic algebra. In the same way as monadic algebras serve as algebraic models of epistemic modal system S5, pseudomonadic algebras serve as algebraic models of doxastic modal system KD45. The main results of the paper are: (1) Characterization of subdirectly irreducible and simple pseudomonadic algebras, as well as Tokarz's proper filter algebras; (2) Order- topological representation of pseudomonadic algebras; (3) Complete description of the lattice of subvarieties of the variety of pseudomonadic algebras.

We introduce a product on an effect algebra. We prove that every product effect algebra with the Riesz decomposition property (RDP), is an interval in an Abelian unital interpolation po-ring, and we show that the category of product effect algebras with the RDP is categorically equivalent with the category of unital Abelian interpolation po-rings. In addition, we show that every product effect algebra with the RDP and with 1 as a product unity is a subdirect product of antilattice product effect algebras with the RDP.

The notion of program-computable function on a universal algebra is formalized through the notion of conditional term. This leads to a somewhat new approach to the study of traditional algebraic problems related to both universal classes and specific finite algebras, and also enables one to pose and investigate the problem of the computation resources of universal algebras.

We investigate general properties of regular irreducible representations of the generalized Heisenberg algebra in an arbitrary nondegenerate weakly closed space with an indefinite metric.

A Kleene-Stone algebra is a bounded distributive lattice with two unary operations that make it a Kleene and a Stone algebra. In this paper, we determine all subdirectly irreducible Kleene-Stone algebras, and describe the free Kleene-Stone algebra on a finite set of generators as a product of certain free Kleene algebras endowed with a Stone negation.

A precise definition of the universal (classical) W algebra for the Wn series is given and its existence and uniqueness are proved. The main observation is that there is a natural reduction from Wn to Wn−1 that allows us to define the universal W algebra as an inverse limit. This universal W algebra is, in a sense, the smallest W algebra from which all Wn can be obtained by reduction. These results extend to other W algebras obtained by reducing the Gel’fand–Dickey brackets, as well as to W superalgebras obtained from the supersymmetric Gel’fand–Dickey brackets.

This is the continuation of the paper “Transformations between Menger systems”. To define when two universal algebras with bases “are the same”, here we propose a universal notion of transformation that comes from a triple characterization concerning three representation facets: the determinations of theHence, this notion consists of three equivalent definitions. It characterizes another technical variant and also the universal version of the very semi-linear transformationsUniversal transformations allow us to check theContrary to present beliefs, even the foundation of abstract Linear Algebra turns out to be incomplete.

We generalize to universal algebra concepts originating fromλ-calculus and programming to prove a new result onthe lattice λTof λ-theories, and ageneral theorem of pure universal algebra which can be seen as ameta version of the Stone Representation Theorem. In this paper weintroduce the class of Church algebras(which includes allBoolean algebras, combinatory algebras, rings with unit and theterm algebras of all λ-theories) to model the"if-then-else" instruction of programming. The interest of Churchalgebras is that each has a Boolean algebra of central elements,which play the role of the idempotent elements in rings. Centralelements are the key tool to represent any Church algebra as a weakBoolean product of indecomposable Church algebras and to prove themeta representation theorem mentioned above. We generalize thenotion of easy λ-term of lambda calculus to provethat any Church algebra with an "easy set" of cardinalitynadmits (at the top) a lattice interval of congruencesisomorphic to the free Boolean algebra with ngenerators.This theorem has the following consequence for the lattice ofλ-theories: for every recursively enumerableλ-theory φand each n, thereis a λ-theory φn? φsuch that {ψ: ψ? φn} "is" the Booleanlattice with 2 nelements.

In this paper we introduce the notion of a

We show that every non-trivial subdirectly irreducible algebra in the variety generated by graph algebras is either a two-element left zero semigroup or a graph algebra itself. We characterize all the subdirectly irreducible algebras in this variety. From this we derive an example of a groupoid (graph algebra) that generates a variety with NP-complete membership problem. This is an improvement over the result of Z. Szekely who constructed an algebra with similar properties in the signature of two binary operations.

Mathematics Technical Report

We discuss formal treatments of composition and terms in universal algebra, propositional logics etc. which may serve as an indispensable base for computer programs capable of term building and term comparison, an important problem in theoretical computer science. After a survey we discuss various Mal'cev algebras introduced for this purpose: preiterative, preiterative with identity, iterative and postiterative. These algebras seem to be simple to implement. We then show how these algebras allow to bring certain universal algebra concepts (as varieties and subvarieties, interpretation and hyperidentities) one conceptual level down. We conclude with a list of properties of preiterative algebras.

This paper investigates endoprimal algebras using techniques from universal algebra but not from duality theory, and thereby exposes quite directly how endoprimality occurs.

An ideal I of a (non-associative) algebra A is dense if the multiplication algebra of A acts faithfully on I, and is complementedly dense if it is a direct summand of a dense ideal. We prove that every complementedly dense ideal of a semiprime algebra is a semiprime algebra, and determine its central closure and its extended centroid. We also prove that a semiprime algebra is an essential subdirect product of prime algebras if and only if, its extended centroid is a direct product of fields. This result is applied to discuss decomposable algebras with respect to some familiar closures for ideals.

An important feature of modem computer algebra systems is the support of a rich type system with the possibility of type inference.

We develop the elementary theory of higher-order universal algebra using the non-standard approach to finite type theory introduced by Henkin. Basic results include: existence theorems for free and initial higher type algebras, a complete higher type equational calculus, and characterisation theorems for higher type equational and Horn classes.