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In 2004, C. Sanza, with the purpose of legitimizing the study of n × m-valued Łukasiewicz algebras with negation (or NSn×m-algebras) introduced 3 × 3-valued Łukasiewicz algebras with negation. Despite the various results obtained about NSn×m-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated NS3×3-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice ∧ (NS3×3) of all subvarieties of NS3×3 and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of NS3×3.

We introduce the definition of the three-element equivalential algebra R with conjunction on the regular elements. We study the variety generated by R and prove the Representation Theorem. Then, we construct the finitely generated free algebras and compute the free spectra in this variety.

In this paper, we expand on the notion of combinatorial presheaf, first introduced explicitly by Aguiar and Mahajan in 2010 but already present in the literature in some other points of view. We do this by adapting the algebraic framework of species to the study of substructures in combinatorics. Afterwards, we consider functions that count the number of patterns of objects and endow the linear span of these functions with a product and a coproduct. In this way, any well-behaved family of combinatorial objects that admits a notion of substructure generates a Hopf algebra, and this association is functorial. For example, the Hopf algebra on permutations studied by Vargas in 2014 and the Hopf algebra on symmetric functions are particular cases of this construction. A specific family of pattern Hopf algebras of interest are the ones arising from commutative combinatorial presheaves. This includes the presheaves on graphs, posets and generalized permutahedra. Here, we show that all the pattern Hopf algebras corresponding to commutative presheaves are free. We also study a remarkable non-commutative presheaf structure on marked permutations, i.e. permutations with a marked element. These objects have a natural product called inflation, which is an operation motivated by factorization theorems for permutations. In this paper, we find new factorization theorems for marked permutations. We use these theorems to show that the pattern Hopf algebra for marked permutations is also free, using Lyndon words techniques. [ABSTRACT FROM AUTHOR]

We extend the concept of “almost indiscernible theory” introduced by Pillay and Sklinos in 2015 (which was itself a modernization and expansion of Baldwin and Shelah from 1983), to uncountable languages and uncountable parameter sequences. Roughly speaking a theory T is almost indiscernible if some saturated model is in the algebraic closure of an indiscernible set of sequences. We show that such a theory T is nonmultidimensional, superstable, and stable in all cardinals ≥ | T | . We prove a structure theorem for sufficiently large a -models M , which states that over a suitable base, M is in the algebraic closure of an independent set of realizations of weight one types (in possibly infinitely many variables). We also explore further the saturated free algebras of Baldwin and Shelah in both the countable and uncountable context. We study in particular theories and varieties of R -modules, characterizing those rings R for which the free R -module on R + generators is saturated, and pointing out a counterexample to a conjecture by Pillay and Sklinos. [ABSTRACT FROM AUTHOR]

We study the existence of free subalgebras in division algebras, and prove the following general result: if A is a noetherian domain which is countably generated over an uncountable algebraically closed field k of characteristic 0, then either the quotient division algebra of A contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if k is an uncountable algebraically closed field and A is a finitely generated k-algebra that is a domain of GK-dimension strictly less than 3, then either A satisfies a polynomial identity, or the quotient division algebra of A contains a free k-algebra on two generators. © 2014 Springer-Verlag Berlin Heidelberg.

In 2004, C. Sanza, with the purpose of legitimizing the study of n x m-valued Lukasiewicz algebras with negation (or NSn x m-algebras) introduced 3 x 3-valued Lukasiewicz algebras with negation. Despite the various results obtained about NSn x m-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a _nite number of free generators. In this note we describe the structure of the free finitely generated N S3 x 3-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice Λ (N S3 x 3) of all subvarieties of N S3 x 3 and we show that the varieties of Boolean algebras, three-valued Lukasiewicz algebras and four-valued Lukasiewicz algebras are proper subvarieties of N S3 x 3. [ABSTRACT FROM AUTHOR]

In this paper we construct 16 free algebras of modular forms on type IV symmetric domains for some reflection groups related to the eight rescaled root lattices A 1 (2) , A 1 (3) , A 1 (4) , 2 A 1 (2) , A 2 (2) , A 2 (3) , A 3 (2) , D 4 (2) . As a corollary, we prove the modularity of formal Fourier–Jacobi expansions for these reflection groups. [ABSTRACT FROM AUTHOR]

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number b n of minimal varieties of given exponent n is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit β = lim n → ∞ b n n exists and can be expressed as the positive solution of an equation a (t) = 0 where a (t) is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank d. It follows from classical results on lacunary power series that the generating function of the sequence b n , n = 1 , 2 , ... , is transcendental. With the same approach we construct examples of free graded semigroups 〈 Y 〉 with the following property. If d n is the number of elements of degree n of 〈 Y 〉 , then the limit δ = lim n → ∞ d n n exists and is transcendental. [ABSTRACT FROM AUTHOR]

In this paper, we study residuated lattices in order to give new characterizations for dense, regular and Boolean elements in residuated lattices and investigate special residuated lattices in order to obtain new characterizations for the directly indecomposable subvariety of Stonean residuated lattices. Free algebra in varieties of Stonean residuated lattices is constructed. We introduce in residuated lattice a new type of filter called special fflter and investigate its properties. Finally, regular fflter property in residuated lattices is introduced and is studied in details. [ABSTRACT FROM AUTHOR]

In 1993, moneyhun showed that if L is a Lie algebra such that dim(L/Z(L)) =n, then dim(L²) ≤ 1/2n(n-1). The author and Saeedi investigated the converse of Moneyhun's result under some conditions .In this paper, We extend their results to obtain several upper bounds for the dimension of a Lie algebra L in terms of dimensions of L², Where L² is the derived subalgebra. Moreover, we give an upper bound for the dimension of the dimension of the c-nilpotent multiplier of a pair of Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper, we show that the derived Picard group of a homologically smooth Koszul connected cochain DG algebra is isomorphic to the opposite group of the derived Picard group of its finite dimensional local Ext-algebra. As applications, we compute the derived Picard groups of some important DG algebras such as trivial DG polynomial algebras, trivial DG free algebras and several non-trivial DG down-up algebras and DG free algebras. [ABSTRACT FROM AUTHOR]

Abstract A variety V is said to be coherent if every finitely generated subalgebra of a finitely presented member of V is finitely presented. It is shown here that coherence corresponds to a key ingredient of uniform deductive interpolation for equational consequence in V : the property that any compact congruence on a finitely generated free algebra of V restricted to a free algebra over fewer generators is compact. A general criterion is derived for establishing failures of coherence, and hence also of uniform deductive interpolation. The criterion is then applied in conjunction with properties of canonical extensions to prove that coherence and uniform deductive interpolation fail for certain varieties of Boolean algebras with operators (including varieties for the modal logic K and KT), double-Heyting algebras, residuated lattices, and lattices. [ABSTRACT FROM AUTHOR]

Celani and Jansana (Math Log Q 58(3):188–207, 2012) give an explicit description of the free implicative semilattice extension of a Hilbert algebra. In this paper, we give an alternative path conducing to this construction. Furthermore, following our procedure, we show that an adjunction can be obtained between the algebraic categories of Hilbert algebras with supremum and that of generalized Heyting algebras. Finally, in the last section, we describe a functor from the algebraic category of Hilbert algebras to that of generalized Heyting algebras, of possible independent interest. [ABSTRACT FROM AUTHOR]

Abstract Arrow logic is a modal logic that is designed to talk about objects that can be illustrated as arrows. In this article, we consider the non-associative arrow logic NAL. We list all the finitely axiomatizable, complete and consistent theories of NAL. This gives an answer to the open problem, posed by I. Németi, addressing the atomicity of the free algebras of the class NA of non-associative relation algebras. We use the method of games as introduced to the fields of logic and algebra by R. Hirsch and I. Hodkinson. We also give a simple proof for the known fact that NAL is decidable through the finite model property. [ABSTRACT FROM AUTHOR]

Abstract An algebra for a functor H is called completely iterative (cia, for short) if every flat recursive equation in it has a unique solution. Every cia is corecursive, i.e., it admits a unique coalgebra-to-algebra morphism from every coalgebra. If the converse also holds, H is called a cia functor. We prove that whenever the base category is hyper-extensive (i.e. countable coproducts are 'well-behaved') and H preserves countable coproducts, then H is a cia functor. Surprisingly few cia functors exist among standard finitary set functors: in fact, the only ones are those preserving coproducts; they are given by X ↦ W × (−) + Y for some sets W and Y. [ABSTRACT FROM AUTHOR]

Abstract Let R be a polynomial ring over a field and I an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension pd (R / I) of R / I is at most 36, although the example with largest projective dimension he constructed has pd (R / I) = 5. Based on computational evidence, it had been conjectured that pd (R / I) ≤ 5. In the present paper we prove this conjectured sharp bound. [ABSTRACT FROM AUTHOR]

Abstract We construct a minimal free resolution of the semigroup ring k [ C ] in terms of minimal resolutions of k [ A ] and k [ B ] when 〈 C 〉 is a numerical semigroup obtained by gluing two numerical semigroups 〈 A 〉 and 〈 B 〉. Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of k [ C ] , and prove that the minimal free resolution of k [ C ] has a differential graded algebra structure provided the resolutions of k [ A ] and k [ B ] possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main result to semigroups in N n. [ABSTRACT FROM AUTHOR]

A variety of linear algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free. A system of elements of a free algebra is primitive if there is a complement of this system with respect to a free generating set of the free algebra. An element of a free algebra of a Schreier variety is said to be almost primitive if it is not primitive in the free algebra, but it is a primitive element of any subalgebra that contains it. This survey article is devoted to the study of primitive and almost primitive elements of Schreier varieties. [ABSTRACT FROM AUTHOR]

This article presents a new method for obtaining small algebras to check the admissibility—equivalently, validity in free algebras—of quasi-identities in a finitely generated quasivariety. Unlike a previous algebraic approach of Metcalfe and Röthlisberger, which is feasible only when the relevant free algebra is not too large, this method exploits natural dualities for quasivarieties to work with structures of smaller cardinality and surjective rather than injective morphisms. A number of case studies are described here that could not be be solved using the algebraic approach, including (quasi)varieties of MS-algebras, double Stone algebras, and involutive Stone algebras. [ABSTRACT FROM AUTHOR]

For a derivation δ of a ring R, we introduce the δ-McCoy rings which are a generalization of the δ-Armendariz rings, and investigate their properties. Some properties of this generalization are established, and connections of properties of a δ-McCoy ring R with n ⨰ n upper triangular T(R, n, σ) are investigated. We study relationship between the δ-McCoy property of R and its polynomial ring, R[x]. We also prove that every ring isomorphism preserves δ-McCoy structure. As a consequence we extend and unify several known results related to McCoy rings. [ABSTRACT FROM AUTHOR]

For a new class of algebras, called EMV-algebras, every idempotent element a determines an MV-algebra which is important for the structure of the EMV-algebra. Therefore, instead of standard homomorphisms of EMV-algebras, we introduce EMV-morphisms as a family of MV-homomorphisms from MV-algebras [0, a] into other ones. EMV-morphisms enable us to study categories of EMV-algebras where objects are EMV-algebras and morphisms are special classes of EMV-morphisms. The category is closed under product. In addition, we define free EMV-algebras on a set X with respect to EMV-morphisms. If X is finite, then a free EMV-algebra on X is termwise equivalent to the free MV-algebra on X. For an infinite set X, the same is true introducing a so-called weakly free EMV-algebra. [ABSTRACT FROM AUTHOR]

Abstract Let α ≥ 2 be any ordinal. We consider the class Drs α 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 Drs α are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class Drs α 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. [ABSTRACT FROM AUTHOR]

It is proved that, for n = 8, 9, 10, the natural algebra of automorphic forms of the group O+2,n(ℤ) acting on the n-dimensional symmetric domain of type IV is free, and the weights of generators are found. This extends results obtained in the author's previous paper for n ≤ 7. On the other hand, as proved in a recent joint paper of the author and O. V. Shvartsman, similar algebras of automorphic forms cannot be free for n > 10. [ABSTRACT FROM AUTHOR]

In this paper a subvariety of m-generalized Łukasiewicz algebras of order n is investigated, particularly that one where m = 2. That is, L2n-algebras are the which ones verifying f4x = x. Because 2 is the smallest number for which Lmn-algebras are not n-valued Łukasiewicz algebras and they constitute a particular case of BCK-algebras, here it is given the whole attention to them. First, some properties of atoms of L2n-algebras are shown, as they provide an important tool to give a complete description of subdirectly irreducible algebras. Next, finite algebras of this variety are studied and it is proved they are direct products of certain subalgebras determined by their atoms. Finally, free L2n-algebras with a finite set of free generators are described. Moreover, the particular cases n = 4 and n = 5 are developed in order to illustrate the applied method. [ABSTRACT FROM AUTHOR]

Let K ⁢ 〈 X 〉 = K ⁢ 〈 X 1 , … , X n 〉 {K\langle X\rangle=K\langle X_{1},\ldots,X_{n}\rangle} be the free algebra generated by X = { X 1 , … , X n } {X=\{X_{1},\ldots,X_{n}\}} over a field K. It is shown that, with respect to any weighted ℕ {\mathbb{N}} -gradation attached to K ⁢ 〈 X 〉 {K\langle X\rangle} , minimal homogeneous generating sets for finitely generated graded two-sided ideals of K ⁢ 〈 X 〉 {K\langle X\rangle} can be algorithmically computed, and that if an ungraded two-sided ideal I of K ⁢ 〈 X 〉 {K\langle X\rangle} has a finite Gröbner basis 𝒢 {{\mathcal{G}}} with respect to a graded monomial ordering on K ⁢ 〈 X 〉 {K\langle X\rangle} , then a minimal standard basis for I can be computed via computing a minimal homogeneous generating set of the associated graded ideal 〈 𝐋𝐇 ⁢ (I) 〉 {\langle\mathbf{LH}(I)\rangle}. [ABSTRACT FROM AUTHOR]

As a proper generalization of the modules with the properties (E) and (EE) that were introduced by Zöschinger in terms of supplements, we say that a module M has the property (WRE) (respectively, (WREE)) if M has a weak Rad-supplement (respectively, ample weak Rad-supplements) in every extension. In this paper, we prove that if every submodule of a module M has the property (WRE), then M has the property (WREE). We show that a ring R is semilocal if and only if every left R-module has the property (WRE). Also we prove that over a commutative Von Neumann regular ring a module M has the property (WRE) if and only if M is injective. [ABSTRACT FROM AUTHOR]

Let L be a relatively free Lie algebra of finite rank n , with n ≥ 2 , L ̂ be the completion of L with respect to the topology defined by the lower central series L = γ 1 (L) ⊇ γ 2 (L) ⊇ ⋯ of L and L (c) = L / γ c (L) , with c ≥ 2. We prove that, with respect to the formal power series topology, the automorphism group Aut (L) of L is dense in the automorphism group Aut (L ̂) of L ̂ if and only if L is nilpotent. Furthermore, we show that there exists a dense subgroup of Aut (L ̂) generated by GL n (K) and a finite set of IA-automorphisms if and only if Aut (L (c)) is generated by GL n (K) and a finite set of IA-automorphisms independent upon c for all c. We apply our study to several varieties of Lie algebras. [ABSTRACT FROM AUTHOR]

Partial words are in our terminology isomorphism classes of labeled partial orders. We consider two structures. Starting with the partial orders reduced to a unique element, the first one is defined as the closure under the three operations of series product, parallel product and ω -power ( ω -series product of the same partial word) and the second one as the closure under the three operations of series product, parallel product and ω -product ( ω -series product of possibly different partial words). We show that the two equational theories can be axiomatized by an infinite collection of simple identities, and that no finite axiomatization exists. We characterize the free algebras in the corresponding varieties as algebras of certain partial words subject to order and graph theoretic conditions. We also show that the first equational theory is decidable. [ABSTRACT FROM AUTHOR]

The commutator [a,b] = ab−ba in a free Zinbiel algebra (dual Leibniz algebra) is an anticommutative operation which satisfies no new relations in arity 3. Dzhumadildaev discovered a relation T(a,b,c,d) which he called the tortkara identity and showed that it implies every relation satisfied by the Zinbiel commutator in arity 4. Kolesnikov constructed examples of anticommutative algebras satisfying T(a,b,c,d) which cannot be embedded into the commutator algebra of a Zinbiel algebra. We consider the tortkara triple product [a,b,c] = [[a,b],c] in a free Zinbiel algebra and use computer algebra to construct a relation TT(a,b,c,d,e) which implies every relation satisfied by [a,b,c] in arity 5. Thus, although tortkara algebras are defined by a cubic binary operad (with no Koszul dual), the corresponding triple systems are defined by a quadratic ternary operad (with a Koszul dual). We use computer algebra to construct a relation in arity 7 satisfied by [a,b,c] which does not follow from the relations of lower arity. It remains an open problem to determine whether there are further new identities in arity n≥9. [ABSTRACT FROM AUTHOR]

Modal pseudocomplemented De Morgan algebras (or mpM-algebras) were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1 (2014), pp. 65{79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying x∧(∼x)* = (∼(x∧(˼x)*))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see:16,17,18. [ABSTRACT FROM AUTHOR]

This survey article illustrates many important current trends and perspectives for the field including classification of hyperidentities, characterizations of algebras with hyperidentities, functional representations of free algebras, structure results for bilattices, categorical questions and applications. However, the paper contains new results and open problems, too. [ABSTRACT FROM AUTHOR]

MMI3-algebras are a generalization of the monadic Tarski algebras as defined by A. Monteiro and L. Iturrioz, and a particular case of the MMIn+1-algebras defined by A. Figallo. They can also be seen as monadic three-valued Łukasiewicz algebras without a first element. By using this point of view, and the free monadic extensions, we construct the free MMI3-algebras on a finite number of generators, and indicate the coordinates of the generators. As a byproduct, we also obtain a construction of the free monadic Tarski algebras.

An algebra S is called a Skolem–Noether algebra (SN algebra for short) if for every central simple algebra R , every homomorphism R → R ⊗ S extends to an inner automorphism of R ⊗ S . One of the important properties of such an algebra is that each automorphism of a matrix algebra over S is the composition of an inner automorphism with an automorphism of S . The bulk of the paper is devoted to finding properties and examples of SN algebras. The classical Skolem–Noether theorem implies that every central simple algebra is SN. In this article it is shown that actually so is every semilocal, and hence every finite-dimensional algebra. Not every domain is SN, but, for instance, unique factorization domains, polynomial algebras and free algebras are. Further, an algebra S is SN if and only if the power series algebra S [ [ ξ ] ] is SN. [ABSTRACT FROM AUTHOR]

A general construction of the free algebra over a poset in varieties finitely generated is given in [8]. In this paper, we apply this to the varieties of Łukasiewicz-Moisil algebras, giving a detailed description of the free algebra over a finite poset (X, ≤), Freen((X, ≤)). As a consequence of this description, the cardinality of Freen((X, ≤)) is computed for special posets. [ABSTRACT FROM AUTHOR]

MMI3-algebras are a generalization of the monadic Tarski algebras as defined by A. Monteiro and L. Iturrioz, and a particular case of the MMIn+1-algebras defined by A. Figallo. They can also be seen as monadic three-valued Łukasiewicz algebras without a first element. By using this point of view, and the free monadic extensions, we construct the free MMI3-algebras on a finite number of generators, and indicate the coordinates of the generators. As a byproduct, we also obtain a construction of the free monadic Tarski algebras. [ABSTRACT FROM AUTHOR]

We show that a monomial idealIin a polynomial ringShas projective dimension ≤ 1 if and only if the minimal free resolution ofS∕Iis supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of theS∕I. We also provide a new characterization of quasi-trees, which we use to give a new proof to a result by Herzog, Hibi, and Zheng which characterizes monomial ideals of projective dimension 1 in terms of quasi-trees. [ABSTRACT FROM PUBLISHER]

We establish the Gröbner-Shirshov bases theory for Lie differential -algebras. As an application, we give a linear basis of a free Lie differential Rota-Baxter algebra on a set. [ABSTRACT FROM AUTHOR]

We introduce the theory of new algebras which have MV-algebra signature enriched with unary operations. These operations are of closure operation type. The appropriate logical system with modalities is constructed. [ABSTRACT FROM AUTHOR]

We establish that the Segre product between a polynomial ring on a field K in m variables and the second squarefree Veronese subalgebra of a polynomial ring on K in n variables has the intersection degree equal to three. We describe a class of monomial ideals of the Segre product with linear quotients. [ABSTRACT FROM AUTHOR]

We present an equivalence between the category of Nelson Paraconsistent lattices (NPc-lattices) and a category of pairs of Brouwerian algebras and regular filters. Specializing such category of pairs to Gödel hoops, we get the subvariety of Gödel NPc-lattices and, using the dual equivalence of finite Gödel hoops with finite trees, we obtain a duality for finite Gödel NPc-lattices. This duality is used to describe finitely generated free Gödel NPc-lattices. [ABSTRACT FROM AUTHOR]

The classes of non-commutative cylindric algebras and weakened cylindric algebras were shown, by Istv'an N'emeti, to have decidable equational theories. In this article, we give new proof for this result and we give an answer to the open problem, posed by N'emetin 1985, addressing the atomicity of the finitely generated free algebras of these classes. We prove that the free algebras of these classes are not atomic. [ABSTRACT FROM AUTHOR]

This paper gives the construction of free medial quandles as well as free n-symmetric medial quandles and free m-reductive medial quandles. [ABSTRACT FROM AUTHOR]

Let $\mathbb{K}$ be a field and S = ${\mathbb{K}}$[x1, . . ., xn] be the polynomial ring in n variables over the field $\mathbb{K}$. For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals. [ABSTRACT FROM PUBLISHER]