Extensions of algebraic systems

Transactions of the American Mathematical Society, 1967

The notions of generic minimum polynomial and generic norm for finitedimensional strictly power-associative algebras were introduced by Professor N. Jacobson in [2], [3], generalizing the notions of principal polynomial and norm for associative algebras. In this paper we extend these concepts to infinite-dimensional algebras; in the process we give a coordinate-free approach to the generic norm. In the first section we review the machinery of the differential calculus in infinite-dimensional spaces, including the definition of rational mappings, the Zariski topology, and differential operators. In the second section we establish the result, which is fundamental in the sequel, that every homogeneous polynomial function can be written uniquely as a product of a finite number of irreducible polynomial functions. We then apply this factorization theory to show that under certain general conditions the factors of an automorphic form (relative to a group of linear transformations) are again automorphic forms. The third and fourth sections are devoted to defining and establishing the basic properties of the generic minimum polynomial and generic norm. A generically algebraic algebra is defined as one which is "uniformly" algebraic in the sense that each element x satisfies a monic polynomial «¡*(A) which varies "continuously" as a function of x: mx(x)=0 for «ix(A) = 2 ml(x)Xi where the coefficients are polynomial functions. The generic minimum polynomial is the polynomial of least degree having these properties, and the generic norm is plus or minus its constant term. Using the factorization theorem the standard properties of the generic norm carry over easily. The fifth section discusses the discriminant of an algebra. An algebra is called unramified if its discriminant is not identically zero. Intuitively, this means that the algebra modulo or radical is separable. Our definition is motivated by [1, p. 105]; it differs from that given in [2]. Several investigations revealed a close connection between the generic norm and forms admitting composition [4], [5], [6], [7], [8]. A general conjecture of Professor R. D. Schäfer was that every norm on a normed algebra was a product of irreducible factors of the generic norm. This was known for finite-dimensional algebras [4] and for infinite-dimensional algebras in certain special cases [5]. In the sixth section of the paper we apply our results to settle the general case.

Journal of Mathematical Sciences, 2000

Annals of Mathematical Logic, 1979

Discussiones Mathematicae - General Algebra and Applications, 2001

It is well known that every congruence regular variety is n-permutable (in the sense of [9]) for some n ≥ 2. For the explicit proof see e.g. [2]. The connections between this n and Mal'cev type characterizations of congruence regularity were studied by G.D. Barbour and J.G. Raftery [1]. The concept of local congruence regularity was introduced in [3]. A common generalization of congruence regularity and local congruence regularity was given in [6] under the name "dual congruence regularity with respect to a unary term g". The natural problem arises what modification of n-permutability is satisfied by dually congruence regular varieties. The aim of this paper is to find out such a modification, to characterize varieties satisfying it by a Mal'cev type condition and to show connections with normally presented varieties (see e.g. [5], [8], [11]). The latter concept was introduced already * This paper is a result of the collaboration of the authors within the framework of the "AktionÖsterreich-Tschechische Republik" (grant No. 26p2 "Local and global congruence properties"). 166 I. Chajda and G. Eigenthaler by J. P lonka under a different term; the names "normal identity" and "normal variety" were firstly used by E. Graczyńska in [8].

Czechoslovak Mathematical Journal, 1986

An algebra Ä with a nullary operation 0 is weakly regular if every two congruence в, Ф on A coincide when ver [0]<9 = [0]^. Varieties of such algebras were characterized by many authors, see or [6] and references therein. It is an interesting problem to find weakly regular algebras in varieties which are not varieties of weakly regular algebras. One can find such attempts e.g. in . An algebra A with a nullary operation 0 has (^-transferable principal congruences (briefly 0-TPC) if for each a, b e A there exists an element с of A such that в[а, b) = = 0(0, c). Varieties of such algebras were characterized in . It is easy to prove that every variety i^ with a nullary operation 0 whose all members have 0-TPC is a variety of wekly regular algebras. There is a natural question: under which con dition on 'f" also the weak regularity of algebras of iT impHes 0-TPC. The aim of this paper is to pick out some broad class of varieties whose members have this property. An algebra A has Principal Compact Congruences if every compact congruence on A is principal, i.e. if for every elements a,-, bie A[i = 1, ..., n) there exist elements a, b of A such that Ö(öi, bi) V ... V 9{a", bn) = e (a, b) in the lattice Con A, Varieties of such algebras were characterized in [9], [8], [7], in the case of permutable varieties also in . This conpcet can be modified in the following way: Definition. Let A be an algebra with a nullary operation 0. A has 0-Principal Compact Congruences if for every elements a^, ..., a"e A there exists an element aeA such that 0(0, ai) V ... V e{0,a") = 0(0, a) in Con A, A variety i^ with a nullary operation 0 has 0-Principal Compact Con gruences if every Ae'f has this property. First we characterize such varieties by a Mal'cev type condition:

2000

New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> make an important such contribution.

2000

We define abelian extensions of algebras in congruence-modular varieties. The theory is sufficiently general that it includes, in a natural way, extensions of R-modules for a ring R. We also define a cohomology theory, which we call clone cohomology, such that the cohomology group in dimension one is the group of equivalence classes of extensions.

Journal of the Australian Mathematical Society, 1970

In this paper we describe a way of representing varieties of algebras by algebras. That is, to each variety of algebras we assign an algebra of a certain type, such that two varieties are rationallv equivalent if and only if the assigned algebras are isomorphic.

Journal of the Australian Mathematical Society, 1999

It is shown that a variety ν has distributive congruence lattices if and only if the intersection of two principal congruence relations is definable by equations involving terms with parameters. The nature of the terms involved then provides a useful classification of congruence distributive varieties. In particular, the classification puts into proper perspective two stronger properties. A variety is said to have the Principal Intersection Property if the intersection of any two principal congruence relations is principal, or the Compact Intersection Property if the intersection of two compact congruence relations is compact. For non-congruence-distributive varieties, it is shown that some useful constuctions are nevertheless possible.

Czechoslovak Mathematical Journal, 2006