Set Of Free Generators Research Articles

On Tarski algebras with a finite set of free generators

In this paper, we describe a method to determine the structure of the Tarski algebra with a finite set of free generators which is different to that given by Iturrioz and Monteiro in [Les algèbres de Tarski avec un nombre fini de générateurs libres, in Informe Técnico, Vol. 37 (Instituto de Matemática de la Universidad Nacional del Sur, Bahía Blanca, 1994)].

ON MAXIMAL SUBGROUPS OF FREE OBJECTS OF CERTAIN COMPLETELY REGULAR SEMIGROUP VARIETIES

By adjusting a method of Kadourek and Polák developed for free semigroups satisfying xr≏ x, we prove that if [Formula: see text] is a periodic group variety, then any maximal subgroup of the free object in the completely regular semigroup variety of the form [Formula: see text] is a relatively free group in [Formula: see text] over a suitable set of free generators. When [Formula: see text] is locally finite, we provide some bounds for the sizes of its finitely generated members.

Freeoids: a semi-abstract view on endomorphism monoids of relatively free algebras

A freeoid over a (normally, infinite) set of variables X is defined to be a pair (W, E), where W is a superset of X, and E is a submonoid of W W containing just one extension of every mapping X → W. For instance, if W is a relatively free algebra over a set of free generators X, then the pair F(W) := (W, End(W)) is a freeoid. In the paper, the kernel equivalence and the range of the transformation F are characterized. Freeoids form a category; it is shown that the transformation F gives rise to a functor from the category of relatively free algebras to the category of freeoids which yields a concrete equivalence of the first category to a full subcategory of the second one. Also, the concept of a model of a freeoid is introduced; the variety generated by a free algebra W is shown to be concretely equivalent to the category of models of F(W). The sets X, W, and the algebras W may generally be many-sorted.

On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima's representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.

A characterization of MV-algebras free over finite distributive lattices

Mundici has recently established a characterization of free finitely generated MV-algebras similar in spirit to the representation of the free Boolean algebra with a countably infinite set of free generators as any Boolean algebra that is countable and atomless. No reference to universal properties is made in either theorem. Our main result is an extension of Mundici’s theorem to the whole class of MV-algebras that are free over some finite distributive lattice.

Primitive elements and automorphisms of free algebras of schreier varieties

In this article, we review results on primitive elements of free algebras of main types of Schreier varieties of algebras. A variety of linear algebras over a field is Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. A system of elements of a free algebra is primitive if it is a subset of some set of free generators of this algebra. We consider free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. We present matrix criteria for systems of elements of elements. Primitive elements distinguish automorphisms: endomorphisms sending primitive elements to primitive elements are automorphisms. We give a series of examples of almost primitive elements (an element of a free algebra is almost primitive if it is not a primitive element of the whole algebra, but it is a primitive element of any proper subalgebra which contains it). We also consider generic elements and Δ-primitive elements.

FREE TERNARY ALGEBRAS

A ternary algebra is a bounded distributive lattice with additonal operations e and ~ that satisfies (a+b)~=a~b~, a~~=a, e≤a+a~, e~= e and 0~=1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.

Determination of Endomorphisms of Free Metabelian Lie Algebras

Answering V. E. Shpil′răin’s question [1, Problem 13.66], E. I. Timoshenko [2] found necessary and sufficient conditions on elements of a free finitely-generated metabelian group G of rank n for every endomorphism of G to be uniquely determined from its action on these elements. Let Mn be a free metabelian Lie algebra with n free generators. The purpose of this article consists in clarifying necessary and sufficient conditions on elements b1, b2, . . . , bm of the Lie algebra Mn for every endomorphism φ of this algebra to be uniquely determined from its actions on these elements, or shortly the conditions for b1, b2, . . . , bm to determine each endomorphism of Mn uniquely. Let K be an arbitrary field and let L be a Lie algebra over K. Denote the universal enveloping algebra for L by UL. It is well known that UL is an integral domain. Denote by F the free Lie algebra over K with the set of free generators {y1, . . . , yn}. The universal enveloping algebra of F is a free associative algebra with the same set of free generators. The ideal generated by F in UF is a free right UF -module with basis y1, . . . , yn. Therefore, each element f ∈ FUF has the unique expansion

Free monadic Tarski algebras

A. Monteiro and L. Iturrioz [7], introduced the notion of monadic Tarski algebras as a generalization of the monadic Boolean algebras considered by P. Halmos [6]. They represent the algebraic version of a fragment of the functional monadic calculus. In this paper we determine the structure of the monadic Tarski algebras with a finite set of free generators.

The Torelli groups for genus 2 and 3 surfaces

THE purpose of this paper is to show that the genus 2 Torelli group is free on infinitely many Dehn twists on separating curves. Moreover, the set of free generators can be identified with the set of splittings of the homology of a genus 2 surface into two subspaces mutually orthogonal and unimodular with respect to the intersection pairing. In addition, it is shown that the third integer homology of the genus 3 Torelli group naturally contains an infinitely generated free abelian group. This is a permutation module for the symplectic modular group. The method used is a study of the period mapping to Siegel space. In Sections 2 and 3 we review some background material. Section 4 shows that the genus 2 Torelli group is free. Section 5 contains additional background material and the result on the third homology. Section 6 gives a homological application.

Criteria for the primitivity of a system of elements of a free metabelian group

We state the criteria for the complementability of an arbitrary system of elements to a set of free generators of a free metabelian group.

The structure of free prosupersolvable groups

Oltikar [9], and Oltikar and Ribes [lo], have shown that the p-Sylow subgroups of a finitely generated prosupersolvable group are finitely generated. The argument is topological and gives no estimate on the number of generators of the subgroups. In Section 1 we prove a theorem on finite groups which gives such a bound for supersolvable groups. The nature of the bound is such that it holds for prosupersolvable groups via a limit argument. Oltikar 191, also showed, using a cohomologicai argument of Gruenberg (p. 164 of [5]), that the p-Sylow subgroups of a free prosupersolvable group are free pro-p. The argument provides no construction of a set of free generators. In Section 2 we construct, for a given set of primes n, and index set, I, a profinite group, F, and subset, D, indexed by I. The construction is by an iterated semidirect product of free pro-p subgroups, F,, on explicitly given generators and with explicitly given actions of F, on the generators of F, for p > q. We show that D is a set of generators for F and that F is prosupersolvable, which, since the bounds of Section 1 are attained, shows the latter are best possible. In Section 3 we establish a mapping property of F relative to D which shows it is “the” free prosupersolvable n-group on D, thus giving a fairly explicit structure theory, including a Hall system, for the latter object, defined by this mapping property. In particular this provides a noncohomological proof of Oltikar’s second result. In Section 4 we use the Hall system to get similar descriptions (iterated semidirect products of known groups with known actions) of certain canonical subgroups of F. In Section 5 we derive some facts about its automorphism group. For basic facts about profinite groups, in particular the notion of (supernatural) order and p-Sylow subgroups, see [ 11, 131. For the notion of free pro-C product and free pro-C group see [ 1, 3, 5, 7, 141. For extending basic facts about supersolvable groups to prosupersolvable see [9, lo]; for the 256 0021-8693/83 $3.00

Finitely generated free tetravalent modal algebras

The purpose of this work is to prove that the tetravalent modal algebra with a finite set of free generators is finite and to determine its cardinal number.

Varieties of associative algebras

Suppose ~-[~] is a free associative algebra over a field of characteristic 0 with countable set of free generators ~I,~ ..... ~R,...; ~[~ The ideal J is called fully characteristic, or a T-ideal, if it is closed under all endomorphisms of the algebra T [I~ Between the T-ideals of ~[I] and the varieties of associative algebras over ~there is one-to-one correspondence that reverses inclusions: the ideal of identities 7"(~'dZ) of the variety ~ is a T-ideal. A variety ~ is called Spechtian if each subvariety (including ~'~ ) is defined by a finite number of identities. A well-known, but still unsolved, problem of the theory of rings is the following: Is every variety of associative algebras Spechtian? In the present paper we prove that certain varieties are Spechtian.

On logical systems with implications and theories of algebras

In paper [8] we concerned ourselves with logical systems with implications and paper [9] was devoted to theories of algebras with relations. The object of the present paper is an investigation of connections existing between logical systems and theories of algebras. Consequently, the present paper may be considered as part 3 of a more exhaustive work, its first two parts being papers [8] and [9]. These three papers are devoted to the same general problem, that is the problem of settling the connections existing between logical systems and mathematical theories. In this work, we first define the relation of "corresponding35 (which occurs between theories of algebras and logical systems) and we study the connections occurring between this relation and the equivalence relation of logical systems or the equivalence relation of theories of algebras. Then we concern ourselves with the problem what properties of logical systems become transposed on the theories of algebras which correspond to these lo? gical systems and vice versa. It follows from Theorem 5 of [9] that classes of models of equivalent theories of algebras only consist of models such that for every model Sffix belonging to one class there exists a model 2Ji2 belonging to the other class such that there exist definable extensions 9Kf and 2Jt* of models 9ft x and 9Jl2, respectively, which are identical. Classes of models of equivalent theories, although may be virtually different, are however similar to the extent of possessing a common class consisting of definable extensions of models of both the first and the second class. The class of all models of a theory T of algebras will be denoted by M (T). Let sft(T) be the algebra of terms of a theory T. Algebra of terms plays a special role in the class of models of theory of algebras. In [9] we proved that two non-degenerated1 theories of algebras such that for their algebras of terms there exist definable exten? sions which are isomorphic, are equivalent in the sense given in [9]. If algebra sft(T) is free in M (T), then the following set a (T) = {[t]t; t is an individual variable}2 is a set of free generators for sft(T). For every logical system with implication there exists its algebra of Lindenbaum. A similar role that is played by algebras of terms for the theories of algebras, is played

Relatively free, nearly nilpotent groups

We show that a free nilpotent group of countable rank, as well as a free group of countable rank of the variety defined by the identity [[x1 x2,..., xn], [xn+1, xn+2]]=1, satisfies the maximal condition for normal subgroups admitting endomorphisms induced by order preserving one-to-one mappings of the set of free generators into itself.