Automorphic Equivalence of Many-Sorted Algebras. Part I (original) (raw)

Automorphisms of the semigroup of endomorphisms of free algebras of homogeneous varieties

2005

We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F be a free algebra of some variety A of linear algebras over K freely generated by a finite set X, EndF be the semigroup of endomorphisms of F, and AutEndF be the group of automorphisms of the semigroup EndF. We investigate structure of the group AutEndF and its relation to the algebraical and categorical equivalence of algebras from A. We define a wide class of R1MF-domains containing, in particular, Bezout domains, unique factorization domains, and some other domains. We show that every automorphism of semigroup EndF, where F is a free finitely generated Lie algebra over an R1MF-domain, is semi-inner. This solves the Problem 5.1 left open in [21]. As a corollary, semi-innerity of all automorphism of the category of free Lie algebras over R1MF-domains is obtained. Relations between categorical and geometrical equivalence of ...

Automorphisms of the endomorphism semigroups of free linear algebras of homogeneous varieties

Linear Algebra and its Applications, 2008

We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F = F (x 1 ,. .. , x n) be a free algebra of some variety of linear algebras over K freely generated by a set X = {x 1 ,. .. , x n }, End F be the semigroup of endomorphisms of F, and Aut End F be the group of automorphisms of the semigroup End F. We investigate the structure of the group Aut End F and its relation to the algebraic and categorical equivalence of algebras from. We define a wide class of R 1 MF-domains containing, in particular, Bezout domains, unique factorization domains, and some other domains. We show that every automorphism of semigroup End F, where F is a free finitely generated Lie algebra over an R 1 MF-domain, is semi-inner. This solves the Problem 5.1 left open in [G. Mashevitzky, B. Plotkin, E. Plotkin, Automorphisms of the category of free Lie algebras, J. Algebra 282 (2004) 490-512]. As a corollary, semi-inner character of all automorphisms of the category of free Lie algebras over R 1 MF-domains is obtained. Relations between categorical and geometrical equivalence of Lie algebras over R 1 MF-domains are clarified. The group Aut End F for the variety of m-nilpotent associative algebras over R 1 MF-domains is described. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of n × n matrices over R 1 MF-domains is obtained. We give an example of the variety of linear algebras over a Dedekind domain such that not all automorphisms of Aut End F are quasi-inner. The results obtained generalize the previous studies of various special cases of varieties of linear algebras over infinite fields.

Automorphisms of categories of free algebras of some varieties

Journal of Algebra, 2006

Let V be a variety of universal algebras. A method is suggested for describing automorphisms of a category of free V-algebras. Applying this general method all automorphisms of such categories are found in two cases: 1) V is the variety of all free associative K−algebras over an infinite field K and 2) V is the variety of all representations of groups in unital R−modules over a commutative associative ring with unit. It turns out that they are almost inner in a sense.

Automorphic equivalence of the representations of Lie algebras

2012

We prove that if a field k is infinite, char(k)=0 and k has not nontrivial automorphisms then automorphic equivalence of representations of Lie algebras coincide with geometric equivalence. We achieve our result by consideration of 1-sorted objects. We suppose that our method can be perspective in the further researches.

AUTOMORPHIC EQUIVALENCE PROBLEM FOR FREE ASSOCIATIVE ALGEBRAS OF RANK TWO

International Journal of Algebra and Computation, 2007

Let K x, y be the free associative algebra of rank 2 over an algebraically closed constructive field of any characteristic. We present an algorithm which decides whether or not two elements in K x, y are equivalent under an automorphism of K x, y .

The Automorphism Group Of Unary Algebras

It was observed by several authors that the congruence lattice Con A and the automorphism group Aut A of a unary algebra are not independent [3]. This paper contains some results concerning this interdependence. For example, if A is a cyclic unary algebra, then S(Aut A), the subgroup lattice of its automorphism group, is isomorphic to an interval of Con A. If A is subalgebra-simple or subdirect irreducible, then S(Aut A) is isomorphic to a principal ideal of Con A. Other results are formulated in terms of fixpoints of automorphisms. For instance, we show that, if there exists an element of A which is not a fixpoint fOFany nontrival automorphism of A, then S(Aut A) can be embedded as a sublattice in ConA. Denote by Inv A the normal subgroup of all automorphisms of A under which all congruences of A are invariant. In general this is quite arbitrary, but we show that for a unary algebra Inv A is either a quaternion group or a subdirect irreducible Abelian group or it is a subdirect product of such groups. If A is cyclic, then Inv A is Hamiltonian. We also prove that if Con A is finite, then so is AutA.

Algebras with actions and automata

International Journal of Mathematics and Mathematical Sciences, 1982

In the present paper we want to give a common structure theory of left action, group operations, R-modules and automata of different types defined over various kinds of carrier objects: sets, graphs, presheaves, sheaves, topological spaces (in particular: compactly generated Hausdorff spaces). The first section gives an axiomatic approach to algebraic structures relative to a base category B, slightly more powerful than that of monadic (tripleable) functors. In section 2 we generalize Lawveres functorial semantics to many-sorted algebras over cartesian closed categories.

Some logical invariants of algebras and logical relations between algebras

St. Petersburg Mathematical Journal, 2008

Let Θ be an arbitrary variety of algebras and H an algebra in Θ. Along with algebraic geometry in Θ over the distinguished algebra H, a logical geometry in Θ over H is considered. This insight leads to a system of notions and stimulates a number of new problems. Some logical invariants of algebras H ∈ Θ are introduced and logical relations between different H 1 and H 2 in Θ are analyzed. The paper contains a brief review of ideas of logical geometry (§1), the necessary material from algebraic logic (§2), and a deeper introduction to the subject (§3). Also, a list of problems is given. 0.1. Introduction. The paper consists of three sections. A reader wishing to get a feeling of the subject and to understand the logic of the main ideas can confine himself to §1. A more advanced look at the topic of the paper is presented in § §2 and 3. In §1 we give a list of the main notions, formulate some results, and specify problems. Not all the notions used in §1 are well formalized and commonly known. In particular, we operate with algebraic logic, referring to §2 for precise definitions. However, §1 is self-contained from the viewpoint of ideas of universal algebraic geometry and logical geometry. Old and new notions from algebraic logic are collected in §2. Here we define the Halmos categories and multisorted Halmos algebras related to a variety Θ of algebras. §3 is a continuation of §1. Here we give necessary proofs and discuss problems. The main problem we are interested in is what are the algebras with the same geometrical logic. The theory described in the paper has deep ties with model theory, and some problems are of a model-theoretic nature. We emphasize once again that §1 gives a complete insight on the subject, while §2 and §3 describe and decode the material of §1. §1. Preliminaries. General view 1.1. Main idea. We fix an arbitrary variety Θ of algebras. Throughout the paper we consider algebras H in Θ. To each algebra H ∈ Θ one can attach an algebraic geometry (AG) in Θ over H and a logical geometry (LG) in Θ over H. In algebraic geometry we consider algebraic sets over H, while in logical geometry we consider logical (elementary) sets over H. These latter sets are related to the elementary logic, i.e., to the first order logic (FOL). Consideration of these sets gives grounds to geometries in an arbitrary variety of algebras. We distinguish algebraic and logical geometries in Θ. However, there is very 2000 Mathematics Subject Classification. Primary 03G25.

On some operations on classes of algebras

Algebra Universalis, 1972

Given any class K of algebras of the same similarity type we let HK and IK denote respectively the classes of all homomorphic and isomorphic images of algebras in K. We also let SK and PK denote respectively the classes of all algebras isomorphic to a subalgebra of a member of K and to a direct product of an arbitrary system of members of K. H, S, P, and I can be conceived of as operations on arbitrary classes of similar algebras and together with all operations constructed from them by composition they are called C-operations. For example, HSP is the operation which at an arbitrary class K takes the value H (S (PK)).