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Problems and results in tame congruence theory  A survey of the '88 Budapest Workshop
 Algebra Universalis
, 1992
"... . Tame congruence theory is a powerful new tool, developed by Ralph McKenzie, to investigate finite algebraic structures. In the summer of 1988, many prominent researchers in this field visited Budapest, Hungary. This paper is a survey of problems and ideas that came up during these visits. It is ..."
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. Tame congruence theory is a powerful new tool, developed by Ralph McKenzie, to investigate finite algebraic structures. In the summer of 1988, many prominent researchers in this field visited Budapest, Hungary. This paper is a survey of problems and ideas that came up during these visits. It is intended both for beginners and experts, who want to do research, or just want to see what is going on, in this new, active area. An Appendix, written in April, 1990, is attached to the paper to summarize new developments. 1. Introduction First we list the names of those colleagues who were so kind as to accept our invitation to the Workshop. They are: Clifford Bergman, Joel Berman, G'abor Cz'edli, Keith Kearnes, George McNulty, Ralph McKenzie, James B. Nation, Peter P'alfy, Robert W. Quackenbush, ' Agnes Szendrei, Matthew Valeriote. Most of them spent only one or two weeks in Budapest, but Joel Berman, ' Agnes Szendrei, and Matthew Valeriote were here for almost a month. This meetin...
EMBEDDING GENERAL ALGEBRAS INTO MODULES
"... The problem of embedding general algebras into modules is revisited. We provide a new method of embedding, based on Jeˇzek’s embedding into semimodules. We obtain several interesting consequences: a simpler syntactic characterization of quasiaffine algebras, a proof that quasiaffine algebras witho ..."
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The problem of embedding general algebras into modules is revisited. We provide a new method of embedding, based on Jeˇzek’s embedding into semimodules. We obtain several interesting consequences: a simpler syntactic characterization of quasiaffine algebras, a proof that quasiaffine algebras without nullary operations are actually quasilinear, and several facts regarding the “abelian iff quasiaffine” problem.
Residual smallness and weak centrality
 The International Journal of Algebra and Computation
"... Abstract. We develop a method of creating skew congruences on subpowers of finite algebras using groups of twin polynomials, and apply it to the investigation of residually small varieties generated by nilpotent algebras. We prove that a residually small variety generated by a finite nilpotent (in p ..."
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Abstract. We develop a method of creating skew congruences on subpowers of finite algebras using groups of twin polynomials, and apply it to the investigation of residually small varieties generated by nilpotent algebras. We prove that a residually small variety generated by a finite nilpotent (in particular, a solvable Eminimal) algebra is weakly abelian. Conversely, we show in two special cases that a weakly abelian variety is residually bounded by a finite number: when it is generated by an Eminimal, or by a finite strongly nilpotent algebra. This establishes the RSconjecture for Eminimal algebras. 1.
FINITE AXIOMATIZABILITY OF CONGRUENCE RICH VARIETIES
"... In this paper we introduce the notion of a congruence rich variety of algebras, and investigate which locally finite subvarieties of such a variety are relatively finitely based. We apply the results obtained to investigate the finite axiomatizability of an interesting variety generated by a particu ..."
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In this paper we introduce the notion of a congruence rich variety of algebras, and investigate which locally finite subvarieties of such a variety are relatively finitely based. We apply the results obtained to investigate the finite axiomatizability of an interesting variety generated by a particular fiveelement directoid.
Chief Factor Sizes in Finitely Generated Varieties
"... Abstract. Let A be a kelement algebra whose chief factor size is c. We show that if B is in the variety generated by A, then any abelian chief factor of B that is not strongly abelian has size at most c k−1. This solves Problem 5 of The Structure of Finite Algebras, by D. Hobby and R. McKenzie. We ..."
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Abstract. Let A be a kelement algebra whose chief factor size is c. We show that if B is in the variety generated by A, then any abelian chief factor of B that is not strongly abelian has size at most c k−1. This solves Problem 5 of The Structure of Finite Algebras, by D. Hobby and R. McKenzie. We refine this bound to c in the situation where the variety generated by A omits type 1. Asageneralization, we bound the size of multitraces of types 1, 2,and3 by extending coordinatization theory. Finally, we exhibit some examples of bad behavior, even in varieties satisfying a congruence identity.
THE WEAK EXTENSION PROPERTY AND FINITE AXIOMATIZABILITY FOR QUASIVARIETIES
"... We define and compare a selection of congruence properties of quasivarieties, including the relative congruence meet semidistributivity, RSD(∧), and the weak extension property, WEP. We prove that if K ⊆ L ⊆ L ′ are quasivarieties of finite signature, and L ′ is finitely generated while K  = WEP, ..."
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We define and compare a selection of congruence properties of quasivarieties, including the relative congruence meet semidistributivity, RSD(∧), and the weak extension property, WEP. We prove that if K ⊆ L ⊆ L ′ are quasivarieties of finite signature, and L ′ is finitely generated while K  = WEP, then K is finitely axiomatizable relative to L. We prove for any quasivariety K that K  = RSD(∧) iff K has pseudocomplemented congruence lattices and K  = WEP. Applying these results and other results proved in M. Maróti, R. McKenzie [17] we prove that a finitely generated quasivariety of finite signature L is finitely axiomatizable provided that L satisfies RSD(∧), or that L is relatively congruence modular and is included in a residually small congruence modular variety. This yields as a corollary the full version of R. Willard’s theorem for quasivarieties and partially proves a conjecture of D. Pigozzi. Finally, we provide a quasiMaltsev type characterization for RSD(∧) quasivarieties and supply an algorithm for recognizing when the quasivariety generated by a finite set of finite algebras satisfies RSD(∧). 1.
ALGORITHMIC PROBLEMS IN VARIETIES
, 1994
"... Mankind always sets itself only such problems as it can solve. Karl Marx, The Introduction to "A Critique of Political Economy". ..."
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Mankind always sets itself only such problems as it can solve. Karl Marx, The Introduction to &quot;A Critique of Political Economy&quot;.
AN OVERVIEW OF MODERN UNIVERSAL ALGEBRA
"... Abstract. This article, aimed specifically at young mathematical logicians, gives a gentle introduction to some of the central achievements and problems of universal algebra during the last 25 years. I intend in this article to introduce nonspecialists to the fact that there are deep results in cont ..."
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Abstract. This article, aimed specifically at young mathematical logicians, gives a gentle introduction to some of the central achievements and problems of universal algebra during the last 25 years. I intend in this article to introduce nonspecialists to the fact that there are deep results in contemporary universal algebra. The first four sections give the context in which universal algebra has something to say, and describe some of the basic results upon which much of the work in the field is built. Section 5 covers the highlights of tame congruence theory, a sophisticated point of view from which to analyze locally finite algebras. Section 6 describes some of the field’s “big ” results and open problems concerning finite algebras, notably the undecidability of certain finite axiomatizability problems and related problems, and the socalled “RS problem, ” currently the most important open problem in the field. This article presents a personal view of current universal algebra, one which is limited both by my ignorance of large parts of the field as well as the need to keep the article focused. For example, I do not mention natural duality theory, one of the
ON THE RELATIONSHIP OF AP, RS AND CEP IN CONGRUENCE MODULAR VARIETIES. II
"... Abstract. Let V be a congruence distributive variety, or a congruence modular variety whose free algebra on 2 generators is finite. If V is residually small and has the amalgamation property, then it has the congruence extension property. Several applications are presented. In two previous papers [1 ..."
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Abstract. Let V be a congruence distributive variety, or a congruence modular variety whose free algebra on 2 generators is finite. If V is residually small and has the amalgamation property, then it has the congruence extension property. Several applications are presented. In two previous papers [1] and [2], we considered the following question: if V is a residually small variety with the amalgamation property, must V have the congruence extension property? Our work established the following implications for a congruence modular variety V: (1) If V is 2–finite and has C2, then AP + RS = ⇒ R. (2) If V is 4–finite with C2 and R, then AP + RS = ⇒ CEP. (The terminology will be explained below.) In this paper we supplement and extend these results. Assuming still that V is congruence modular, we have: (3) AP + RS = ⇒ C2. (4) If V has R, then AP + RS = ⇒ CEP. Combining these implications, we have that every congruence modular, 2–finite variety satisfies AP + RS = ⇒ CEP. Furthermore, every congruence distributive variety (no finiteness assumption) satisfies AP + RS = ⇒ CEP. Our universal algebraic notation and terminology are standard. Good references are [4] and [9]. Let V be a variety of algebras. We say that V • has the amalgamation property (AP) if, for all A, B0, B1 ∈Vand all embeddings fi: A → Bi, fori=0,1, there is C ∈V and embeddings gi: Bi → C, i =0,1, such that g0 ◦ f0 = g1 ◦ f1, • is residually small (RS) if there is a cardinal κ such that every subdirectly irreducible algebra in V has cardinality less than κ, • has the congruence extension property (CEP) if, for all A ≤ B ∈V,and congruence α on A, thereis¯α∈Con B such that ¯α ↾ A = α, • is n–finite, for a positive integer n, ifeverymemberofVgenerated by n elements is finite.