Free Algebra

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We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free * -algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.

Let T be a submonad of the ultrafilter monad β and let G be a subfunctor of the filter functor. The T-algebras are topological spaces whose closed sets are the subalgebras and form thereby an equationally definable full subcategory of topological spaces. For appropriate T , countably generated free algebras provide ZFC examples of separable, Urysohn, countably compact, countably tight spaces which are neither compact nor sequential, and 2 c non-homeomorphic such examples exist. For any space X, say that U ⊂ X is G-open if U belongs to every ultrafilter in G X which converges in U . The full subcategory Top G consists of all G-spaces, those spaces in which every G-open set is open. Each Top G has at least these stability properties: it contains all Alexandroff spaces, and is closed under coproducts, quotients and locally closed subspaces. Examples include sequential spaces, P -spaces and countably tight spaces. T-algebras are characterized as the T -compact, T -Hausdorff T -spaces. Malyhin's theorem on countable tightness generalizes verbatim to Top G for any G ⊂ β. For r ∈ ω = βω\ω, let G r be the subfunctor of β generated by r and let T r be the generated submonad. If RK is the Rudin-Keisler preorder on ω , r RK s ⇔ G r ⊂ G s . Let c be the Comfort preorder and define the monadic preorder r m s to mean T r ⊂ T s . Then r RK s ⇒ r m s ⇒ r c s. It follows that there exist 2 c monadic types. For each such type T r , the T r -algebras form an equationally definable full subcategory of topological spaces with only one operation of countably infinite arity. No two of these varieties are term equivalent nor is any one a full subcategory of another inside topological spaces. Say that r ∈ ω is an m-point if G r = T r . Under CH, m-points exist.

We consider a couple of versions of classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra?) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a variety of algebras of this signature such that the free algebra of the variety contains multilinear elements of arbitrary large degree, while the clone of every such element satisfies some nontrivial identity. If, in addition, the number of binary operations is at least 2, then one can guarantee that each such clone is finitely-dimensional.

Conformal Geometric Algebraic (CGA) provides ideal mathematical tools for construction, analysis and integration of classical Euclidean, Inversive & Projective Geometries, with practical applications to computer science, engineering and physics. This paper is a comprehensive introduction to a CGA tool kit. Synthetic statements in classical geometry translate directly to coordinate-free algebraic forms. Invariant and covariant methods are coordinated by conformal splits, which are readily related to the literature using methods of matrix algebra, biquaternions and screw theory. Designs for a complete system of powerful tools for the mechanics of linked rigid bodies are presented.

Nagata gave a fundamental sucient condition on group actions on nitely generated commutative algebras for nite generation of the subalge- bra of invariants. In this paper we consider groups acting on noncommutative algebras over a eld of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corre- sponding relatively

Let α ≥ 2 be any ordinal. We consider the class Drsα of relativized diagonal free set algebras of dimension α. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of Drsα are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class Drsα corresponds to first order logic, without equality symbol, with α-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.

For algebras A whose type is given by an endofunctor, iterativity means that every flat equation morphism in A has a unique solution. In our previous work we proved that every object generates a free iterative algebra, and we provided a coalgebraic construction of those free algebras. Iterativity w.r.t. an endofunctor was generalized by Tarmo Uustalu to iterativity w.r.t. a "base", i.e., a functor of two variables yielding finitary monads in one variable. In the current paper we introduce iterative algebras in this general setting, and provide again a coalgebraic construction of free iterative algebras.

Motivated by recent work of Connes and Marcolli, based on the Connes-Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on properties of Hopf algebras encapsulated in the notion of associated descent algebras. Besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures, the new techniques give rise to an algebraic approach to the Galois theory of renormalization. In particular, they do not depend on the geometry underlying the case of dimensional regularization and the Riemann-Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme.

A logic is developed in which function symbols are allowed to represent partial functions. It has the usual rules of logic (in the form of a sequent calculus) except that the substitution rule has to be modified. It is developed here in its minimal form, with equality and conjunction, as "partial Horn logic".

In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Groebner-Shirshov bases of free Rota-Baxter algebra, $\lambda$-differential algebra and $\lambda$-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to those obtained by Ebrahimi-Fard and Guo, and Guo

A Σ-labeled n-poset is an (at most) countable set, labeled in the set Σ, equipped with n partial orders. The collection of all Σ-labeled n-posets is naturally equipped with n binary product operations and n ω-ary product operations. Moreover, the ω-ary product operations give rise to n ω-power operations. We show that those Σ-labeled n-posets that can be generated from the singletons by the binary and ω-ary product operations form the free algebra on Σ in a variety axiomatizable by an infinite collection of simple equations. When n = 1, this variety coincides with the class of ω-semigroups of Perrin and Pin. Moreover, we show that those Σ-labeled n-posets that can be generated from the singletons by the binary product operations and the ω-power operations form the free algebra on Σ in a related variety that generalizes Wilke's algebras. We also give graph-theoretic characterizations of those n-posets contained in the above free algebras. Our results serve as a preliminary study to a development of a theory of higher dimensional automata and languages on infinitary associative structures.

We study z-automorphisms of the polynomial algebra K[x, y, z] and the free associative algebra K 〈x, y, z〉 over a field K, i.e., automorphisms that fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K 〈x, y, z〉 we include also results about the structure of the z-tame automorphisms and algorithms that recognize z-tame automorphisms and z-tame coordinates.

This paper shows that the collection of identities which hold in the algebra N of the natural numbers with constant zero, and binary operations of sum and maximum is not finitely based. Moreover, it is proven that, for every n, the equations in at most n variables that hold in N do not form an equational basis. As a stepping stone in the proof of these facts, several results of independent interest are obtained. In particular, explicit descriptions of the free algebras in the variety generated by N are offered. Such descriptions are based upon a geometric characterization of the equations that hold in N, which also yields that the equational theory of N is decidable in exponential time.

Let α ≥ 2 be any ordinal. We consider the class Drsα of relativized diagonal free set algebras of dimension α. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of Drsα are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class Drsα corresponds to first order logic, without equality symbol, with α-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.

Completely iterative algebras (cias) are those algebras in which recursive equations have unique solutions. In this paper we study complete iterativity for algebras with computational effects (described by a monad). First, we prove that for every analytic endofunctor on Set there exists a canonical distributive law over any commutative monad M , hence a lifting of that endofunctor to the Kleisli category of M. Then, for an arbitrary distributive law λ of an endofunctor H on Set over a monad M we introduce λ-cias. The cias for the corresponding lifting of H (called Kleisli-cias) form a full subcategory of the category of λ-cias. For various monads of interest we prove that free Kleisli-cias coincide with free λ-cias, and these free algebras are given by free algebras for H. Finally, for three concrete examples of monads we prove that Kleisli-cias and λ-cias coincide and give a characterisation of those algebras.

We consider the lambda definability problem over an arbitrary free algebra. There is a natural notion of primitive recursive function in such algebras and at the same time a natural notion of a lambda definable function. We shown that the question: ”For a given free algebra and a primitive recursive function within this algebra decide whether this function is lambda definable” is undecidable if the algebra is infinite. The main part of the paper is dedicated to the algebra of numbers in which lambda definability is described by the Schwichtenberg theorem. The result for an arbitrary infinite free algebra has been obtained by a simple interpretation of numerical functions as recursive functions in this algebra. This result is a counterpart of the distinguished result of Loader in which lambda terms are evaluated in finite domains for which undecidability of lambda definability is proved.

Using ideas of our recent w ork on automorphisms of residually nilpotent relatively free groups, we i n troduce a new growth function for subgroups of the automorphism groups of relatively free algebras FnV o ver a eld of characteristic zero and the related notion of Gelfand Kirillov dimension, and study their behavior. We prove that, under some natural restrictions, the Gelfand Kirillov dimension of the group of tame automorphisms of FnV is equal to the Gelfand Kirillov dimension of the algebra FnV. We show that, in some cases, the Gelfand Kirillov dimension of the group of tame automorphisms of FnV i s smaller than the Gelfand Kirillov dimension of the whole automorphism group, and calculate the Gelfand Kirillov dimension of the automorphism group of FnV for some important v arieties V.

We study different properties of the Nagata automorphism of the polynomial algebra in three variables and extend them to other automorphisms of polynomial algebras and algebras close to them. In particular, we propose two approaches to the Nagata conjecture: via the theory of Griibner bases and trying to lift the Nagata automorphism to an automorphism of the free associative algebra. We show that the reduced Grijbner basis of three face polynomials of the Nagata automorphism obtained by substituting a variable by zero does not produce an automorphism, independently of the "tag" monomial ordering, contrary to the two variable case. We construct examples related to Nagata's automorphism which show different aspects of this problem. We formulate a conjecture which implies Nagata's conjecture. We also construct an explicit lifting of the Nagata automorphism to the free metabelian associative algebra. Finally, we show that the method to determine whether an endomorphism of K[X] is an automorphism is based on a general fact for the ideals of arbitrary free algebras and works also for other algebraic systems such as groups and semigroups, etc.

In this paper we describe completely the involutions of the first kind of the algebra UT n (F ) of n × n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT n (F ) when n is even and a single class in the odd case.

Let A=A(x_1,...,x_n) be a free associative algebra in A freely generated over K by a set X={x_1,...,x_n}, End A be the semigroup of endomorphisms of A, and Aut End A be the group of automorphisms of the semigroup End A. We investigate the structure of the groups Aut End A and Aut A^∘, where A^∘ is the category of finitely generated free algebras from A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End F and the group Aut A^∘ is generated by semi-inner and mirror automorphisms of the category A^∘. This result solves an open Problem formulated in 22

Residuated logic is a generalization of intuitionistic logic, which does not assume the idempotence of the conjunction operator, Such generalized conjunction operators have proved important in expert systems (in the area of Approximate Reasoning) and in some areas of Theoretical Computer Science. Here we generalize the intuitionistic tableau procedure and prove that this generalized tableau method is sound for the semantics (the class of residuated algebras) of residuated propositional calculus (RPC). Since the axioms of RPC are complete for the semantics we may conclude that whenever a formula ~b is tableau provable, it is deducible in RPC. We present two different approaches for constructing residuated algebras which give us countermodels for some formulas • which are not tableau provable, The first uses the fact that the theory of residuated algebras is equational, to construct quotients of free algebras. The second uses finite algebras. We end by discussing a number of open questions.

We consider algebras for which the operation PC of pure closure of subsets satisfies the exchange property. Subsets that are independent with respect to PC are directly independent. We investigate algebras in which PC satisfies the exchange property and which are relatively free on a directly independent generating subset. Examples of such algebras include independence algebras and dinitely generated free modules over principal ideal domains.

We introduce a semantical definition of minterms and maxterms which generalizes the usual notion in Boolean logic to a class of many-valued logics. We apply this notion to get normal forms for logics G, NM, NMG. Then we obtain a combinatorial description of the n-generated free algebras in the varieties constituting the algebraic semantics of those logics. Specifically, we represent via combinatorial posets the embedding of the n-generated free algebra into the direct product of all n-generated chains in the variety.

This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max, +) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

Modal transition systems (MTS) is a formalism which extends the classical notion of labelled transition systems by introducing transitions of two types: must transitions that have to be present in any implementation of the MTS and may transitions that are allowed but not required.

We study z-automorphisms of the polynomial algebra K[x, y, z] and the free associative algebra K x, y, z over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K x, y, z we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms and z-tame coordinates.

be the polynomial algebra in two variables over the finite field F q with q elements. We give an exact formula and the asymptotics for the number p n of automorphisms (f, g) of F q [x, y] such that max{deg(f ), deg(g)} = n. We describe also the Dirichlet series generating function The same results hold for the automorphisms of the free associative algebra F q x, y . We have also obtained analogues for free algebras with two generators in Nielsen -Schreier varieties of algebras. 2000 Mathematics Subject Classification. 11T06; 13B25; 05A15; 30B50; 16S10; 17A50.

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 ...

Abstract. Let A = A(x1,...,xn) be a free associative algebra in the variety of associative algebras A freely generated over K by a set X = {x1,...,xn}, End A be the semigroup of endomorphisms of A, and Aut EndA be the group of automorphisms of the semigroup EndA. We investigate the structure of the groups Aut EndA and Aut A ◦ , where A ◦ is the category of finitely generated free algebras from A. We prove that the group Aut EndA is generated by semiinner and mirror automorphisms of EndF and the group Aut A ◦ is generated by semi-inner and mirror automorphisms of the category A ◦. This result solves an open Problem formulated in [14]. 1.

Using actions of the free monoids and free associative algebras, we establish some Schreier-type formulas involving the ranks of actions and the ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish the generalization of the Schreier formula to the case of subgroups of infinite index. We also study and apply large modules over free associative algebras in the spirit of [1,11].

For arbitrary F-algebra, in which the operation of addition is defined, I explore biring of matrices of mappings. The sum of matrices is determined by the sum in F-algebra, and the product of matrices is determined by the product of mappings. The system of equations, whose matrix is a matrix of mappings, is called a system of additive equations. I considered the methods of solving system of additive equations. As an example, I consider the solution of a system of linear equations over the complex field provided that the equations contain unknown quantities and their conjugates. Linear mappings of algebra over a commutative ring preserve the operation of addition in algebra and the product of elements of the algebra by elements of the ring. The representation of tensor product A\otimes A in algebra A generates the set of linear transformations of algebra A. The results of this research will be useful for mathematicians and physicists who deal with different algebras.

We present a survey on methods to analyse the program complexity, based on termination orderings and quasi-interpretations. This method can be implemented to certify the runtime (or space) of programs. We demonstrate that the class of functions computed by first order functional programs over free algebras which terminate by Permutation Path Order-ing (resp. Lexicographic Path Ordering) and admit a quasi-interpretation bounded by a polynomial, is exactly the class of functions computable in polynomial time (resp. space).

We solve a problem of J6nsson by showing that the class Y/of (isomorphs of) algebras of binary relations, under the operations of relative product, conversion, and intersection, and with the identity element as a distinguished constant, is not axiomatizable by a set of equations. We also show that the set of equations valid in ~ is decidable, and in fact the set of equations true in the class of all positive algebras of relations is decidable.

Let A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded. Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One of the main tools of independent interest is the construction in the free non-associative algebra of multialternating polynomials satisfying special properties. addresses: [email protected] (A. Giambruno), [email protected] (I. Shestakov), [email protected] (M. Zaicev).