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Chapter 12 of "The Structure of Finite Algebras" by D. Hobby and R. McKenzie contains theorems revealing how the set of types appearing in a locally finite variety [Formula: see text] influences the size of the free algebra in [Formula: see text] freely generated by n elements. We provide more results in this vein. If A is a subdirectly irreducible algebra of size k, then a lower bound on the number of n-ary polynomials of A is obtained for each case that the monolith of A has type 3, 4, or 5. Examples for every k show that in each case the lower bound is the best possible. As an application of these results we show that for every finite k if all k-element simple algebras are partitioned into five classes according to their type, then algebras in each class have a sharply determined band of possible values for their free spectra. These five bands are disjoint except for some overlap on simple algebras of types 2 and 5.

For the algebra of skew-derivations over a torus, we classify the irreducible weight modules with some restrictions.

Let V be a finitely decidable variety and A ∈ V a finite subdirectly irreducible algebra with a type 2 monolith μ. We prove that (1) the solvable radical ν of A is the centralizer of μ (2) ν is abelian, i.e., every solvable congruence of A is abelian; (3) the interval sublattice I[ν, 1A] ⊆ Con A is linear, and typ{ν, aA} ∈ {3}.

The two-dimensional Euclidean quantum algebra is considered at roots of unity. The algebraic properties and the Poisson-Hopf structure are first investigated. We then determine the irreducible representations and study the reduction of the tensor product of two of them.

Distributive bounded lattices with a dual homomorphism as unary operation, called Ockham algebras, were firstly studied by Berman (1977). The varieties of Boolean algebras, De Morgan algebras, Kleene algebras and Stone algebras are some of the well known subvarieties of Ockham algebra. In this paper, new results about the congruence lattice of Ockham algebras are given. From these results and Urquhart's representation theorem for Ockham algebras a complete characterization of the subdirectly irreducible Ockham algebras is obtained. These results are particularized for a large number of subvarieties of Ockham algebras. For these subvarieties a full description of their subdirectly irreducible algebras is given as well.

Finite‐dimensional irreducible representations of a reflection algebra related to the reflection equation for the su(1,1)q model are classified. It is proven that this algebra has two inequivalent one‐dimensional and one two‐dimensional irreducible representations. Some indecomposable representations are also constructed and the (q‐boson–)fermion realizations are explicitly presented. Finally, the Yang–Baxterization of this algebra and its constant solutions (the one‐dimensional representations) is investigated.

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.

In the present paper we introduce the concept of an L -restrictecd fu11 subdirect product of algebras. This notion is a common generalization of full subdirect and weak direct product. The main result of our paper of a uniqueness theorem for restricted full subdirect representations of algebras. This result implies some theorems on weak direct and full subdirect representations.

In this paper, we define a logical algebra named MP-algebra and discuss its algebraic properties. We find that MP-algebra not only takes Boole algebra, MV-algebra and R 0 algebra as its special examples but also holds the subdirect representation theorem same as that of on Boole algebra, MV algebra and R 0 algebra. We also explore the basic properties of implication operation of MP-algebra. We prove that an MP-algebra is also a residuated lattice with many good properties. The conclusions we got show that MP-algebra is a well-structure logic algebra when it is taken as the logic truth degree set.

We study the Weyl algebra A pertaining to a particle constrained on a sphere, which is generated by the coordinates n and by the angular momentum J . A is the algebra E3 of the Euclidean group in space. We find its irreducible representations by a novel approach, by showing that they are the irreducible representations (l0, 0) of so(3, 1) , with l0 or -l0 being equal to the Casimir operator J.n . Any integer or half-integer l0 is allowed. The Hilbert space of a particle of spin S hosts 2S + 1 such representations. J can be analyzed into the sum L + S , i.e. pure spin states can be identified, provided 2S + 1 irreducible representations of A are glued together. These results apply to any surface which is diffeomorphic to S2.

A semigroupS is said to be right commutative ifaxy=ayx for alla,x,y ∈ S. The object of this paper is to determine the subdirectly irreducible right commutative semigroups.

We prove lemmas in Andrunakievich s-varieties on the transitivity of the relation “is an ideal of” and concerning subdirectly irreducible factor rings. Applying these lemmas we show that a Plotkin radical introduced in [8] has the ADS-property and is ideal hereditary. These lemmas are applicable in proving a subdirect decomposition for rings having an ideal with 0 antisimple radical. For Jordan algebras and near-rings (they do not form Andrunakievich varieties) we can prove a similar subdirect decomposition concerning ideals with 0 Brown-McCoy radical.

0. Introduction. Besides being of intrinsic interest, cylindric (semi-) lattices arise naturally from the study of dependencies in relational databases; the polynomials on a cylindric semilattice are closely related to the queries obtainable from project-join mappings on a relational database (cf. [D] for references). This note is intended to initiate the study of these structures, and only a few, rather basic results will be given. Some problems at the end will hopefully stimulate further research. Related issues are discussed in [H], and for further background material the reader is invited to consult [N]. I should like to thank H. Andreka and I. Nemeti for stimulating discussions on the subject.

The purpose of this paper is to define and investigate the new class of quasi-Stone algebras (QSA's). Among other things we characterize the class of simple QSA's and the class of subdirectly irreducible QSA's. It follows from this characterization that the subdirectly irreducible QSA's form an elementary class and that the variety of QSA's is locally finite. Furthermore we prove that the lattice of subvarieties of QSA's is an (ω + 1)-chain. MSC: 03G25, 06D16, 06E15.

A permutational automaton is a fibered automaton with one-element event set or an empty state space. This paper characterizes all subdirectly irreducible permutational automata.

The present paper is based on [11], where a number of conjectures are made concerning the structure of the lattice of normal extensions of the tense logicKt. That paper was mainly dealing with splittings of and some sublattices, and this is what I will concentrate on here as well. The main tool in analysing the splittings of will be the splitting theorem of [8]. In [11] it was conjectured that each finite subdirectly irreducible algebra splits the lattice of normal extensions ofK4t andS4t. We will show that this is not the case and that on the contrary only very few and trivial splittings of the mentioned lattices exist.

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.

The explicit forms of the irreducible representation matricea for the quantum SLq(3) enveloping algebra are computed in analogy to the irreducible tensor technique in the classical SU(3) algebra.

We construct new types of algebras which take into account the block structure of finite groups. We study the construction of such algebras. It is proved that the number of irreducible components of such an algebra is equal to the number of p blocks of the finite group whose defective groups contain a given p-element defined by the algebra. If the p-element is the unit, then the number of irreducible components is equal to the number of p-blocks of the finite group.

In the long run, mathematics evolves organically and is independent of individual mathematicians. Multiple, and in many cases, almost simultaneous discovery of concepts or proofs of theorems is the norm rather than the exception. Someone will discover the concept or prove the theorem, eventually. Gian-Carlo Rota often praised an idea, detached from its accidental discoverers, as “an idea whose time has come”. In the 1960s, the time came for the theory of Mobius functions of partially ordered sets, and the area has flourished since then. Rota did not include his paper Foundations I [4] among his most original papers. In his opinion, he only uncovered a theory already “ there”.

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.

The left spectrum of a wide class of the algebras of skew differential operators is described. As a sequence, we determine and classify all the algebraically irreducible representations of the quantum Heisenberg algebra over an arbitrary field.

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 prove that every finite, simple, surjective algebra having no proper subalgebras is either quasiprimal or affine or isomorphic to an algebra term equivalent to a matrix power of a unary permutational algebra. Consequently, it generates a minimal variety if and only if it is quasiprimal. We show also that a locally finite, minimal variety omitting type 1 is minimal as a quasivariety if and only if it has a unique subdirectly irreducible algebra.

An application of the theory of Noetherian rings is used to produce a new family of irreducible representations for the Virasoro algebra.

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.

We characterize the simple and subdirectly irreducible distributive algebras in some varieties of distributive lattices with unary operators, including topological and monadic positive modal algebras. Finally, for some varieties of Heyting algebras with operators we apply these results to determine the simple and subdirectly irreducible algebras.

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

Ježek and Kepka [4] proved that a universal algebra A with at least one at least binary operation is isomorphic to the factor of a subdirectly irreducible algebra B by its monolith if and only if the intersection of all of its (nonempty) ideals is nonempty, and that B may be chosen to be finite if A is finite. (By an ideal of A is meant a non-empty subset I of A such that f(a1, . . . , an) ∈ I whenever f is an n-ary fundamental operation of A and a1, . . . , an ∈ A are elements with ai ∈ I for at least one index i.) In the present paper, we prove that if A is a semigroup, then B may be chosen also to be a semigroup, but that a finite semigroup need not be isomorphic to the factor of a finite subdirectly irreducible semigroup by its monolith.

We determine the minimally irreducible groups with proper non-soluble socle, whose degree is the product of two primes.

A generalization to symmetric and k-symmetric algebras of the Bialinicky-Birula representation theorems for De Morgan algebras is proved. All the subdirectly irreducible k-symmetric algebras are characterized. >