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Let F be a field of characteristic zero. By a ⁎-superalgebra we mean an algebra A with graded involution over F. Recently, algebras with graded involution have been extensively studied in PI-theory and the sequence of ⁎-graded codimensions { c n gri (A) } n ≥ 1 has been investigated by several authors. In this paper, we classify varieties generated by unitary ⁎-superalgebras having quadratic growth of ⁎-graded codimensions. As a result we obtain that a unitary ⁎-superalgebra with quadratic growth is T 2 ⁎ -equivalent to a finite direct sum of minimal unitary ⁎-superalgebras with at most quadratic growth, where at least one ⁎-superalgebra of this sum has quadratic growth. Furthermore, we provide a method to determine explicitly the factors of those direct sums. [ABSTRACT FROM AUTHOR]

For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero and let m ≥ 2 be an integer. In this paper, we prove that if a group grading on U T m (F) admits a graded involution then this grading is a coarsening of a Z ⌊ m 2 ⌋ -grading on U T m (F) and the graded involution is equivalent to the reflection or symplectic involution on U T m (F) , this grading is called the finest grading on U T m (F). Furthermore, if m ≤ 4 the algebra U T m (F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z ⌊ m 2 ⌋ , ⁎) -identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z ⌊ m 2 ⌋ , ⁎) -codimensions is determined. As a consequence, we prove that for any G -grading on U T m (F) and any graded involution the (G , ⁎) -exponent is m. Finally, we study the algebra U T 3 (F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z -grading and the Z 2 -grading induced by (0 , 1 , 0). We determine a basis for the (Z 2 , ⁎) -identities and we compute the codimension sequence for the (Z 2 , ⁎) -graded identities for U T 3 (F). [ABSTRACT FROM AUTHOR]

In a previous paper [4] we tried to build the basic theory of unbounded Tomita's observable algebras called T †-algebras which are related to unbounded operator algebras, especially unbounded Tomita-Takesaki theory, operator algebras on Krein spaces, studies of positive linear functionals on *-algebras and so on. And we defined the notions of regularity, semisimplicity and singularity of T †-algebras and characterized them. In this paper we shall proceed further with our studies of T †-algebras and investigate whether a T †-algebra is decomposable into a regular part and a singular part. [ABSTRACT FROM AUTHOR]

This paper enriches a pre-existing decision algorithm, which in turn augmented a fragment of Tarski's elementary algebra with one-argument real functions endowed with a continuous first derivative. In its present (still quantifier-free) version, our decidable language embodies the addition of functions and multiplication of functions by scalars; the issue we address is the one of satisfiability. As regards real numbers, individual variables and constructs designating the basic arithmetic operations are available, along with comparison relators. As regards functions, we have variables of another sort, out of which compound terms are formed by means of constructs designating addition and differentiation. An array of predicates designates various relationships between functions, as well as function properties, that may hold over intervals of the real line; those are: function comparisons, strict and non-strict monotonicity / convexity / concavity, comparisons between the derivative of a function and a real-valued term. Our decision method consists in preprocessing the given formula into an equi-satisfiable quantifier-free formula of the elementary algebra of real numbers, whose satisfiability can then be checked by means of Tarski's decision method. No direct reference to functions will appear in the target formula, each function variable having been superseded by a collection of stub real variables; hence, in order to prove that the proposed translation is satisfiability-preserving, we must figure out a flexible-enough family of interpolating C 1 functions that can accommodate a model for the source formula whenever the target formula turns out to be satisfiable. With respect to the results announced in earlier papers of the same stream, a significant effort went into designing the family of interpolating functions so that it could meet the new constraints stemming from the presence of function addition (along with differentiation) among the constructs of our fragment of mathematical analysis. • A formal language RDF* concerned with differentiable real functions is proposed. • The class of differentiable functions treated is closed under addition. • The expressive power of RDF* is illustrated through a gallery of examples. • A satisfiability-preserving algorithm reducing RDF* formulas into an existential sentence of Tarskian algebra is presented. • The correctness of the reduction algorithm is reported. [ABSTRACT FROM AUTHOR]

In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra A h generated by elements x , y , which satisfy the relation y x − x y = h for some 0 ≠ h ∈ F [ x ]. In this paper we investigate the standard polynomial identities and minimal identities for certain subspaces of A h over an infinite field of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we will provide a condition for a Z 2 -grading on E to behave like the natural Z 2 -grading E c a n. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [10]. The conjecture poses that every Z 2 -grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of E c a n by means of its Z 2 -graded polynomial identities. Furthermore we construct a Z 2 -grading on E that gives a negative answer to the conjecture. [ABSTRACT FROM AUTHOR]

Let n be an integer greater than 1, and M n (C) be the algebra of all n × n -complex matrices. Let x 0 ∈ C n be a nonzero vector, and Φ be a linear map on M n (C) such that Φ (I) is invertible. For any matrix T ∈ M n (C) , let γ (T , x 0) denote the local reduced minimum modulus of T at x 0. In this paper, we show that Φ satisfies γ (T , x 0) = 0 ⇔ γ (Φ (T) , x 0) = 0 , (T ∈ M n (C)) , if and only if there are two invertible matrices A , B ∈ M n (C) such that A x 0 = A ⁎ x 0 = x 0 and Φ (T) = B T A for all T ∈ M n (C). When n = 2 , we show that the invertibility hypothesis of Φ (I) is redundant. [ABSTRACT FROM AUTHOR]

Max-plus algebra is an algebraic structure in which classical addition and multiplication are replaced by maximum and addition, respectively. We say that the columns of a real matrix A are strongly independent if the max-plus linear system A ⊗ x = b has a unique solution for at least one real vector b. A square matrix A with strongly independent columns is called strongly regular. The investigation of the properties of regularity is important for applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. The present paper studies three versions of the regularity of matrices and interval matrices, namely, strong regularity, von Neumann regularity and Gondran-Minoux regularity. For each concept of regularity we will present equivalent conditions which can be verified in polynomial time. [ABSTRACT FROM AUTHOR]

In geometric tolerancing, functional specification is becoming challenging as the complex mutual situations of geometrical features increase and the requisites for complex assembly accuracy escalate. A computational representation of sets of situation features is proposed in this paper to manage functional specification of complex assemblies. A minimum triplet set of a point, a straight line and a plane called ToLiP, is introduced to represent situation features. Functional specification operations can thus be mathematically established using projective geometric algebra (PGA). The effectiveness of the proposed approach is highlighted through a case study using quadrupole magnets of a particle accelerator. [ABSTRACT FROM AUTHOR]

In this paper we study identities and images of polynomials on null-filiform Leibniz algebras. If L n is an n -dimensional null-filiform Leibniz algebra, we exhibit a finite minimal basis for Id (L n) , the polynomial identities of L n , and we explicitly compute the images of multihomogeneous polynomials on L n. We present necessary and sufficient conditions for the image of a multihomogeneous polynomial f to be a subspace of L n. For the particular case of multilinear polynomials, we prove that the image is always a vector space, showing that the analogue of the L'vov-Kaplansky conjecture holds for L n. We also prove similar results for an analog of null-filiform Leibniz algebras in the infinite-dimensional case. [ABSTRACT FROM AUTHOR]

The problem of discretization of Darboux integrable equations is considered. Given a Darboux integrable continuous equation, one can obtain a Darboux integrable differential-discrete equation, using the integrals of the continuous equation. In the present paper, the discretization of the differential-discrete equations is done using the corresponding characteristic algebras. New examples of integrable discrete equations are obtained. [ABSTRACT FROM AUTHOR]

The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-complex supporting minimal free resolution of the ideal and its Artinian reduction. Also, in this case, we compute the multigraded Betti numbers, Cohen-Macaulay type and determine when the corresponding algebra is a level algebra. [ABSTRACT FROM AUTHOR]

Let G be a group and F an infinite field. Assume that A is a finite dimensional F -algebra with an elementary G -grading. In this paper, we study the graded identities satisfied by the tensor product grading on the F -algebra A ⊗ C , where C is an H -graded colour β -commutative algebra. More precisely, under a technical condition, we provide a basis for the T G -ideal of graded polynomial identities of A ⊗ C , up to graded monomial identities. Furthermore, the F -algebra of upper block-triangular matrices U T (d 1 , ... , d n) , as well as the matrix algebra M n (F) , with an elementary grading such that the neutral component corresponds to its diagonal, are studied. As a consequence of our results, a basis for the graded identities, up to graded monomial identities of degrees ≤ 2 d − 1 , for M d (E) and M q (F) ⊗ U T (d 1 , ... , d n) , with a tensor product grading, is exhibited. In this latter case, d = d 1 + ... + d n. Here E denotes the infinite dimensional Grassmann algebra with its natural Z 2 -grading, and the grading on M q (F) is Pauli grading. The results presented in this paper generalize results from [14] and from other papers which were obtained for fields of characteristic zero. [ABSTRACT FROM AUTHOR]

A universal algebra A with underlying set A is said to be a matroid algebra if (A , 〈 ⋅ 〉) , where 〈 ⋅ 〉 denotes the operator subalgebra generated by , is a matroid. A matroid algebra is said to be an independence algebra if every mapping α : X → A defined on a minimal generating X of A can be extended to an endomorphism of A. These algebras are particularly well-behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics, such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well-defined notion of dimension. Let A be any independence algebra of finite dimension n , with at least two elements. Denote by End (A) the monoid of endomorphisms of A. In the 1970s, Głazek proposed the problem of extending the matrix theory for vector spaces to a class of universal algebras which included independence algebras. In this paper, we answer that problem by developing a theory of matrices for (almost all) finite-dimensional independence algebras. In the process of solving this, we explain the relation between the classification of independence algebras obtained by Urbanik in the 1960s, and the classification of finite independence algebras up to endomorphism-equivalence obtained by Cameron and Szabó in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szabó to all independence algebras. The paper closes with a number of questions for experts on matrix theory, groups, semigroups, universal algebra, set theory or model theory. [ABSTRACT FROM AUTHOR]

The n -th Schrödinger algebra sch n defined in [14] is the semi-direct product of the Lie algebra sl 2 with the n -th Heisenberg Lie algebra h n , which generalizes the Schrödinger algebra sl 2 ⋉ h 1. Let ϕ : h n → C be a nonzero Lie algebra homomorphism. A sch n -module V is called quasi-Whittaker of type ϕ if V = U (sch n) v , where U (sch n) is the universal enveloping algebra of sch n , v is a nonzero vector such that x v = ϕ (x) v for any x ∈ h n. In this paper, we prove that a simple sch n -module V is a quasi-Whittaker module if and only if V is a locally finite h n -module. Then we classify the simple quasi-Whittaker modules of ϕ , according to the rank of ϕ. Furthermore, we characterize arbitrary quasi-Whittaker modules through the rank of ϕ. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let c n L (A) , n ≥ 1 , be its differential codimension sequence. Such sequence is exponentially bounded and exp L ⁡ (A) = lim n → ∞ ⁡ c n L (A) n is an integer that can be computed, called differential PI-exponent of A. In this paper we prove that for any Lie algebra L , exp L ⁡ (A) coincides with exp ⁡ (A) , the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L -algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth. [ABSTRACT FROM AUTHOR]

Let A be a unital C⁎-algebra equipped with its natural order. As usual the effect algebra of A is the interval { a ∈ A : 0 ≤ a ≤ I A } , where I A denotes the unit of A. In this paper, we give a complete description of order isomorphisms between effect algebras of C⁎-algebras of type C (X) ⊗ B (H) , where C (X) stands for the algebra of all continuous complex valued functions on a (non pathological) Hausdorff compact space X and B (H) denotes the algebra of all bounded linear operators on a complex Hilbert space H. Our results generalize some works by L. Molnár and P. Šemrl. [ABSTRACT FROM AUTHOR]

In this paper we construct a graded universal enveloping algebra of a G -graded Lie algebra, where G is not necessarily an abelian group. If the grading group is abelian, then it coincides with the classical construction. We prove the existence and uniqueness of the graded enveloping algebra. As consequences, we prove a graded variant of Witt's Theorem on the universal enveloping algebra of the free Lie algebra, and the graded version of Ado's Theorem, which states that every finite-dimensional Lie algebra admits a faithful finite dimensional representation. Furthermore we investigate if a Lie grading is equivalent to an abelian grading. [ABSTRACT FROM AUTHOR]

Let M be a type II 1 von Neumann algebra, S (M) be the Murray–von Neumann algebra associated with M and let A be a ⁎-subalgebra in S (M) with M ⊆ A. We prove that any ring derivation D from A into S (M) is necessarily inner. Further, we prove that if A is an E W ⁎ -algebra such that its bounded part A b is a W ⁎ -algebra without finite type I direct summands, then any ring derivation D from A into L S (A b) — the algebra of all locally measurable operators affiliated with A b , is an inner derivation. We also give an example showing that the condition M ⊆ A is essential. At the end of this paper, we provide several criteria for an abelian extended W ⁎ -algebra such that all ring derivations on it are linear. [ABSTRACT FROM AUTHOR]

This paper introduces a logical characterization of multi-adjoint algebras with a twofold contribution. On the one hand, the study of multi-adjoint algebras, from a logical perspective, will allow us to discover both the core and new features of these algebras. On the other hand, the axiomatization of multi-adjoint algebras will be useful to take advantage of the properties of the logical connectives considered in the corresponding deductive system. The mechanism considered to carry out the mentioned axiomatization follows the one given by Petr Hájek for residuated lattices. Specifically, the paper presents the bounded poset logic (BPL) as an axiomatization of a bounded poset, since this algebraic structure is the most simple structure from which a multi-adjoint algebra is defined. In the following, the language of BPL is enriched with a family of pairs, composed of a conjunctor and an implication, and its axiomatic system is endowed with new axioms, giving rise to the multi-adjoint logic (ML). The soundness and completeness of BPL and ML are proven. Finally, a comparison between the axiomatization of the multi-adjoint logic and the one given for the BL logic is introduced. [ABSTRACT FROM AUTHOR]

Let B (X) be the algebra of all bounded linear operators on a complex or real Banach space X with dim ⁡ X ≥ 3. In this paper, we characterize the maps from B (X) into itself which preserve the ascent of product of operators or, they preserve the descent of product of operators. It turns out that both problems are connected with preservers of the rank-one nilpotency of the product. [ABSTRACT FROM AUTHOR]

The present paper introduces a new definition of entropy on sequential effect algebras. Unlike the logarithmic behaviour of the entropy defined in the literature, the entropy considered here is of exponential nature. Conditional entropy and entropy on the dynamical system are also introduced and studied. It is also proved that entropy of the dynamical system is invariant under isomorphism. [ABSTRACT FROM AUTHOR]

For a frame Q , this paper studies the Kleisli monoids and Eilenberg-Moore algebras with respect to the Q -Scott open set monad. Firstly, we show that the Q -Scott open set functor can form a monad, and prove that the Kleisli monoids of this monad are exactly the algebraic Q -ordered closure spaces or equivalently the strong Q -convex spaces. Then we show that the Eilenberg-Moore algebras of Q -Scott open set monad are precisely the algebraic Q -modules or equivalently the fuzzy frames. Finally, we construct the strong Q -Scott open set monad, and study the corresponding Kleisli monoids and the Eilenberg-Moore algebras. [ABSTRACT FROM AUTHOR]

Let A be a unital C ⁎ algebra and for every a ∈ A , r (a) denote the numerical radius of a ∈ A. The power inequality for numerical radius states that for every polynomial P (z) = z n and a ∈ A the inequality P (r (a)) ≥ r (P (a)) holds. In this paper, we get a characterization of polynomials with real coefficients that satisfy the power inequality on all 2 × 2 matrices with real entries. We also characterize all polynomials that satisfy the power inequality on every commutative unital C ⁎ algebra. [ABSTRACT FROM AUTHOR]

An algebra 〈 A , ∧ , ∨ , → , □ , 0 , 1 〉 of type (2 , 2 , 2 , 1 , 0 , 0) is said to be a modal weak Heyting algebra if 〈 A , ∧ , ∨ , → , 0 , 1 〉 is a weak Heyting algebra and the following conditions are satisfied for every a , b ∈ A : M1) □ (1) = 1 , M2) □ (a ∧ b) = □ (a) ∧ □ (b) and M3) □ (a → b) ≤ □ (a) → □ (b). If this algebra satisfies the inequality a ∧ (a → b) ≤ b then it is called modal RWH-algebra. In this paper we study the variety of modal RWH-algebras, which is denoted by KRWH, and some of its subvarieties. We focus our attention on the study of the lattice of congruences of any member of KRWH and some related properties. In particular, we give an equational basis for the subvariety of KRWH generated by the class of their totally ordered members. [ABSTRACT FROM AUTHOR]

In this paper we study the lattice of restricted subalgebras of a restricted Lie algebra. In particular, we consider those algebras in which this lattice is dually atomistic, lower or upper semimodular, or in which every restricted subalgebra is a quasi-ideal. The fact that there are one-dimensional subalgebras which are not restricted results in some of these conditions being weaker than for the corresponding conditions in the non-restricted case. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalization of the classical Skjelbred–Sund method, used to classify nilpotent low-dimensional Lie algebras, in order to classify Lie-Yamaguti algebras with non-trivial annihilator. We develop this method with the purpose of classifying nilpotent Lie-Yamaguti algebras, and we obtain from it the algebraic classification of the nilpotent Lie-Yamaguti algebras up to dimension four. [ABSTRACT FROM AUTHOR]

Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and Gödel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over N. It is also shown that the existence property (or existential definability property) holds for statements of the form ∃ y σ φ (y) , where the variable y ranges over objects of finite type σ. This applies in particular to CZF (Constructive Zermelo-Fraenkel set theory) and IZF (Intuitionistic Zermelo-Fraenkel set theory), two systems known not to have the general existence property. On the technical side, the paper uses a method that amalgamates generic realizability for set theory with truth, whereby the underlying partial combinatory algebra is required to contain all objects of finite type. [ABSTRACT FROM AUTHOR]

In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update. [ABSTRACT FROM AUTHOR]

Given a unital algebra A of linear transformations on a finite-dimensional complex vector space V , in this paper we study the set Col (A) consisting of those invertible linear transformations S on V which map every subspace M ∈ Lat (A) to a subspace S M ∈ Lat (A). We show that Col (A) is the normalizer of the group of invertible linear transformations in the reflexive cover of A. For the unital algebra (A) which is generated by a linear transformation A , we give the complete description of Col (A). By using the primary decomposition of A , we first reduce the problem of characterizing Col (A) to the problem of characterizing the group Col (N) of a given nilpotent linear transformation N. While Col (N) always contains all invertible linear transformations of the commutant (N) ′ of N , it is always contained in the reflexive cover of (N) ′. We prove that Col (N) is a proper subgroup of (Alg Lat (N) ′) − 1 if and only if at least two Jordan blocks in the Jordan decomposition of N are of dimension 2 or more. We also determine the group Col (J 2 ⊕ J 2). [ABSTRACT FROM AUTHOR]

Although there have been many fuzzy or probabilistic relational database models proposed for representing and handling imprecise and uncertain information of objects in real-world applications, models combining the relevance and strength of both fuzzy set theory and probability theory appear sporadic. In this paper, we propose a new fuzzy and probabilistic relational database model where the imprecision of an attribute value is represented by a fuzzy set and the uncertainty of a relational tuple is represented by a probability interval. The mass assignment theory is employed to deal with the challenge of integration and computation of both fuzzy sets and probabilities in the same model. The conjunction and disjunction strategies to combine imprecise and uncertain information are introduced. Then the fundamental concepts of the classical relational database model are extended and generalized in this new model. The syntax and semantics of the selection operation are formally defined. Finally, the other important algebraic operations on imprecise attributes and uncertain tuples are developed. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study the algebra of almost strongly fuzzy bounded operators on fuzzy normed linear spaces. Some connections with the algebra of strongly fuzzy bounded operators are presented. The completeness of these algebras is also established. A characterization of bounded elements of the algebra of almost strongly fuzzy bounded operators is given. We also introduce two fuzzy spectral radii for almost strongly fuzzy bounded operators. For bounded elements of the considered algebra the equality between these fuzzy spectral radii is proved. [ABSTRACT FROM AUTHOR]

In this paper, the notions of regularity and normality of (L , M) -fuzzy topological spaces are introduced by using residual implication, where L and M are completely distributive De Morgan algebras. It is shown that (L , M) -fuzzy interior operator and (L , M) -fuzzy closure operator can be used to characterize regularity and normality. The relationships among separation axioms of an (L , M) -fuzzy topological space are discussed. Moreover, it is proved that the four separation axioms are equivalent to one another in an (L , M) -fuzzy metric space. [ABSTRACT FROM AUTHOR]

In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong ([20, Theorem 5.1]) and T. Kobayashi ([30, Theorem 1.3]) concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, Arf local rings contain only finitely many Ulrich ideals once the normalization is a local ring. [ABSTRACT FROM AUTHOR]

In this paper, first we give the notion of a compatible 3-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible 3-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible 3-Lie algebra. Then we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible 3-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible 3-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible 3-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible 3-Lie algebra using the second cohomology group. [ABSTRACT FROM AUTHOR]

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local "spin variables" can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent "dual" formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra U q (s l ˆ (2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new integrable hierarchy by using charged free boson and Friedan–Martinec–Shenker (FMS) bosonization, called modified bosonic integrable hierarchy, which is an anologue of the classical (l − l ′) –th modified KP hierarchy. Firstly, the representation of g l ˆ ∞ is given by using charged free bosons. Then the modified bosonic integrable hierarchy is proposed. Next after discussing FMS bosonizations in the language of lattice vertex algebra, bilinear equations of modified bosonic integrable hierarchy are constructed by FMS bosonization, which are rewritten into Hirota bilinear equations by using Hirota derivatives. At last, we consider solutions of this modified bosonic integrable hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of relative and absolute Ω-Rota-Baxter algebras of weight λ , which can be considered as a family algebraic generalization of relative and absolute Rota-Baxter algebras of weight λ. We study the deformations of relative and absolute Ω-Rota-Baxter algebras of arbitrary weight. Explicitly, we construct an L ∞ [ 1 ] -algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative Ω-Rota-Baxter algebra structures of weight λ. For a relative Ω-Rota-Baxter algebra of weight λ , the corresponding twisted L ∞ [ 1 ] -algebra controls its deformations, which leads to the cohomology theory of it, and this cohomology theory can interpret the formal deformations of the relative Ω-Rota-Baxter algebra. Moreover, we also obtain the corresponding results for absolute Ω-Rota-Baxter algebras of weight λ from the relative version. [ABSTRACT FROM AUTHOR]

In this paper, we address the computation of the symmetries of polynomial (and thus also rational) planar vector fields using elements from Computer Algebra. We show that they can be recovered from the symmetries of the roots of an associated univariate complex polynomial which is constructed as a generator of a certain elimination ideal. Computing symmetries of the roots of the auxiliary polynomial is a task considerably simpler than the original problem, which can be done efficiently working with classical Computer Algebra tools. Special cases, in which the group of symmetries of the polynomial roots is infinite, are separately considered and investigated. The presented theory is complemented by illustrative examples. The main steps of the procedure for investigating the symmetries of a given polynomial vector field are summarized in a flow chart for clarity. • Computation of the symmetries of polynomial/rational planar vector fields using Computer Algebra is addressed. • These symmetries can be recovered from the symmetries of the roots of an associated univariate complex polynomial. • Special cases, in which the symmetry group of the polynomial roots is infinite, are separately considered and investigated. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new invariant for a non-degenerate evolution algebra, which consists of an ordered sequence of evolution algebras of lower dimension, belonging all of them to a specific family. We use this invariant to propose a method to classify non-degenerate evolution algebras, and we apply it up to dimension 3. We also use it to describe the derivations of some families of evolution algebras and the degenerations of the variety of evolution algebras with square not greater than 1. [ABSTRACT FROM AUTHOR]

Given a finite dimensional algebra A over a field k , and a finite acyclic quiver Q , let Λ = A ⊗ k k Q / I , where kQ is the path algebra of Q over k and I is a monomial ideal. This paper is devoted to studying the separated monic representations over the bounded quiver (Q , I) over a subcategory X of A -mod, denoted by smon (Q , I , X). We show that (X , Y) is a (complete) hereditary cotorsion pair in A -mod if and only if (smon (Q , I , X) , rep (Q , I , Y)) is a (complete) hereditary cotorsion pair in Λ-mod. We also show that A is left weakly Gorenstein if and only if so is Λ. Provided that k Q / I is non-semisimple, the category ▪ of semi-Gorenstein-projective Λ-modules coincides with the category of separated monic representations ▪ if and only if A is left weakly Gorenstein. [ABSTRACT FROM AUTHOR]

Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yield algebra retracts on the level of cycle algebras. In particular, that allows us to use a powerful algebraic machinery. As an application, we study highest possible degrees in minimal homogeneous systems of generators of defining ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian subgraph polytopes. [ABSTRACT FROM AUTHOR]

In the paper, we investigate a one-to-one correspondence between n -dimensional observables and n -dimensional spectral resolutions with values in a kind of a lexicographic form of quantum structures like perfect MV-algebras or perfect effect algebras. The multidimensional version of this problem is more complicated than a one-dimensional one because if our algebraic structure is k -perfect for k > 1 , then even for the two-dimensional case we have more characteristic points. The obtained results are also applied to existence of an n -dimensional meet joint observable of n one-dimensional observables on a perfect MV-algebra. The results are divided into two parts. In Part I, we present notions of n -dimensional observables and n -dimensional spectral resolutions with accent on lexicographic type effect algebras and lexicographic MV-algebras. We concentrate on characteristic points of spectral resolutions and the main body is in Part II where one-to-one relations between observables and spectral resolutions are presented. [ABSTRACT FROM AUTHOR]

In this paper, the well-known Faulkner construction is revisited and adapted to include the super case, which gives a bijective correspondence between generalized Jordan (super)pairs and faithful Lie (super)algebra (super)modules, under certain constraints (bilinear forms with properties analogous to the ones of a Killing form are required, and only finite-dimensional objects are considered). We always assume that the base field has characteristic different from 2. It is also proven that associated objects in this Faulkner correspondence have isomorphic automorphism group schemes. Finally, this correspondence will be used to transfer the construction of the tensor product to the class of generalized Jordan (super)pairs with "good" bilinear forms. [ABSTRACT FROM AUTHOR]

Let us consider a parametric weighted directed graph in which every arc (j , i) has weight of the form w ((j , i)) = max (P i j + λ , I i j − λ , C i j) , where λ is a real parameter and P , I and C are arbitrary square matrices with elements in R ∪ { − ∞ }. In this paper, we design an algorithm that solves the Non-positive Circuit weight Problem (NCP) on this class of parametric graphs, which consists in finding all values of λ such that the graph does not contain circuits with positive weight. This problem, which generalizes other instances of the NCP previously investigated in the literature, has applications in the consistency analysis of a class of discrete-event systems called P-time event graphs. The proposed algorithm is based on max-plus algebra and formal languages, and improves the worst-case complexity of other existing approaches, achieving strongly polynomial time complexity O (n 4) (where n is the number of nodes in the graph). [ABSTRACT FROM AUTHOR]