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We consider the moment map m : ℙ V n → i (n) for the action of GL (n) on V n = ⊗ 2 (ℂ n) ∗ ⊗ ℂ n , and study the functional F n = ∥ m ∥ 2 restricted to the projectivizations of the algebraic varieties of all n -dimensional Leibniz algebras L n and all n -dimensional symmetric Leibniz algebras S n , respectively. First, we give a description of the maxima and minima of the functional F n : L n → ℝ , proving that they are actually attained at the symmetric Leibniz algebras. Then, for an arbitrary critical point [ μ ] of F n : S n → ℝ , we characterize the structure of [ μ ] by virtue of the nonnegative rationality. Finally, we classify the critical points of F n : S n → ℝ for n = 2 , 3 , respectively. [ABSTRACT FROM AUTHOR]

We construct a finite-dimensional metabelian right-symmetric algebra over an arbitrary field that does not have a finite basis of identities. [ABSTRACT FROM AUTHOR]

Let F be an algebraically closed field of characteristic zero and G a cyclic group of odd prime order. We consider the class of finite dimensional (G , ⁎) -algebras, namely G -graded algebras endowed with graded involution ⁎, and we characterize the varieties generated by algebras of this class which are minimal with respect to the (G , ⁎) -exponent. [ABSTRACT FROM AUTHOR]

We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realization spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which an n-tuple of collinear points can be lifted to a nondegenerate realization of a point-line configuration. We show that forest configurations are liftable and characterize the realization space of liftable configurations as the solution set of certain linear systems of equations. Moreover, we study the Zariski closure of the realization spaces of liftable and quasi-liftable configurations, known as matroid varieties, and establish their irreducibility. Additionally, we compute an irreducible decomposition for their corresponding circuit varieties. Applying these liftability properties, we present a procedure to generate some of the defining equations of the associated matroid varieties. As corollaries, we provide a geometric representation for the defining equations of two specific examples: the quadrilateral set and the 3 × 4 grid. While the polynomials for the latter were previously computed using specialized algorithms tailored for this configuration, the geometric interpretation of these generators was missing. We compute a minimal generating set for the corresponding ideals. [ABSTRACT FROM AUTHOR]

We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras UT_2(W_\lambda) or End(W_\mu) for some integral dominant weight \lambda,\mu with \mu \neq 0. In the special case L=\mathfrak {sl}_2 we prove that this is a sufficient condition too. [ABSTRACT FROM AUTHOR]

We exhibit a class C of finite dimensional algebras with superinvolution over an algebraically closed field of characteristic zero, with the remarkable property that each member of C generates a minimal variety of algebras with superinvolution. This sums up to the fact that any affine minimal variety of algebras with superinvolution is generated by a suitable member of C , thus providing a complete characterization of the affine minimal varieties of algebras with superinvolution. [ABSTRACT FROM AUTHOR]

We are considering the commuting variety of the Lie algebra pgl n over an algebraically closed field of characteristic p > 0 , namely the set of pairs { (A , B) ∈ pgl n × pgl n | [ A , B ] = 0 }. We prove that if n = p r , then there are precisely two irreducible components, of dimensions n 2 + r − 1 and n 2 + n − 2. We also prove that the variety { (x , y) ∈ G L n (k) × G L n (k) | [ x , y ] = ζ I } is irreducible of dimension n 2 + n / d , where ζ is a root of unity of order d with d dividing n. [ABSTRACT FROM AUTHOR]

Let X ⊂ P (w 0 , w 1 , w 2 , w 3) be a quasismooth well-formed weighted projective hypersurface and let L = lcm (w 0 , w 1 , w 2 , w 3). We characterize when X is rational under the assumption that L divides deg ⁡ (X). Furthermore, we give a new family of normal rational weighted projective hypersurfaces with ample canonical divisor, valid in all dimensions, adding to the list of examples discovered by Kollár. Finally, we determine precisely which affine Pham-Brieskorn threefolds are rational, answering a question of Rajendra V. Gurjar. [ABSTRACT FROM AUTHOR]

We construct free algebras in the variety generated by the equivalential algebra with conjunction on dense elements and compute the formula for the free spectrum of this variety. Moreover, we describe the decomposition of free algebras into directly indecomposable factors. [ABSTRACT FROM AUTHOR]

The goal of this paper is to describe the group of strongly stable automorphisms for categories of free finitely generated non-associative algebras with unit for classical varieties of non-associative algebras, in particular for the varieties of commutative, Jordan, alternative and power associative algebras. The motivation of this research is tightly connected with some problems in Universal Algebraic Geometry. [ABSTRACT FROM AUTHOR]

In this survey paper, we collect properties of one-relator algebras in varieties of linear algebras. The main attention is paid to Schreier varieties of algebras. A variety of linear algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free. [ABSTRACT FROM AUTHOR]

We study minimal and almost minimal varieties of commutative integral residuated lattices. In particular, we prove that the variety G B A of generalized Boolean algebras and the variety C generated by the negative integers are the only two semilinear atoms. Furthermore, we give a characterization of all the finitely generated covers of G B A , showing that there are infinitely many, we axiomatize the unique common cover of G B A and C , and we construct continuum-many semilinear covers of C and infinitely many non-semilinear ones. [ABSTRACT FROM AUTHOR]

We prove that if, for a non‐trivial variety of non‐associative algebras, every subalgebra of every free algebra is free and I2$I^2$ is an ideal whenever I$I$ is an ideal, then this variety coincides with the variety of all Lie algebras. [ABSTRACT FROM AUTHOR]

We study many-valued coalgebraic logics with semi-primal algebras of truthdegrees. We provide a systematic way to lift endofunctors defined on the variety of Boolean algebras to endofunctors on the variety generated by a semi-primal algebra. We show that this can be extended to a technique to lift classical coalgebraic logics to many-valued ones, and that (one-step) completeness and expressivity are preserved under this lifting. For specific classes of endofunctors, we also describe how to obtain an axiomatization of the lifted many-valued logic directly from an axiomatization of the original classical one. In particular, we apply all of these techniques to classical modal logic. [ABSTRACT FROM AUTHOR]

Motivated by Haviar and Ploščica's 2021 characterisation of Boolean products of simple De Morgan algebras, we investigate Boolean products of simple algebras in filtral varieties. We provide two main theorems. The first yields Werner's Boolean-product representation of algebras in a discriminator variety as an immediate application. The second, which applies to algebras in which the top congruence is compact, yields a generalisation of the Haviar–Ploščica result to semisimple varieties of Ockham algebras. The property of having factor principal congruences is fundamental to both theorems. While major parts of our general theorems can be derived from results in the literature, we offer new, self-contained and essentially elementary proofs. [ABSTRACT FROM AUTHOR]

The representation of elements of free non-associative algebras as a set of multidimensional tables of coefficients is defined. An operation for finding partial derivatives for elements of free non-associative algebras in the same form is considered. Using this representation, a criterion of primitivity for elements of lengths 2 and 3 in terms of matrix ranks, as well as a primitivity test for elements of arbitrary length, is derived. This test makes it possible to estimate the number of primitive elements in free non-associative algebras with two generators over a finite field. The proposed representation allows us to optimize algorithms for symbolic computations with primitive elements. Using these algorithms, we find the number of primitive elements of length 4 in a free non-associative algebra of rank 2 over a finite field. [ABSTRACT FROM AUTHOR]

We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localization, analogous results to those obtained in the category of groups hold, and we provide existence theorems for certain localization functors in specific semi-abelian categories. We prove that a Birkhoff subcategory of an ideal determined category yields a conditionally flat localization, and explain how conditional flatness corresponds to the property of admissibility of an adjunction from the point of view of categorical Galois theory. Under the assumption of fiberwise localization, we give a simple criterion to determine when a (normal epi)-reflection is a torsion-free reflection. This is shown to apply, in particular, to nullification functors in any semi-abelian variety of universal algebras. We also relate semi-left-exactness for a localization functor L with what is called right properness for the L -local model structure. [ABSTRACT FROM AUTHOR]

Let K be a finite field of characteristic 2, and U T 2 : = U T 2 (K) be the Lie algebra of 2 × 2 upper triangular matrices over K with the multiplication x ∘ y = x y + y x = x y − y x. In this paper, we exhibit a finite basis of graded identities for the variety of Lie algebras generated by U T 2 for any grading and show that it has the Specht property. It is important to highlight that the technique used in order to solve the Specht problem is independent of the characteristic of the field and also of its cardinality. [ABSTRACT FROM AUTHOR]

We investigate in this article regular Heyting algebras by means of Esakia duality. In particular, we give a characterisation of Esakia spaces dual to regular Heyting algebras and we show that there are continuum-many varieties of Heyting algebras generated by regular Heyting algebras. We also study several logical applications of these classes of objects and we use them to provide novel topological completeness theorems for inquisitive logic, DNA -logics and dependence logic. [ABSTRACT FROM AUTHOR]

Based on the notion of full transformations with fixed set, in this paper, we present a novel concept of n-ary Fix(In,Y)-full terms. This term can be considered as a generalization of strongly full terms, permutational full terms and full terms. Together with the superposition operation, one can form a Menger algebra of rank n. The freeness of such algebra with respect to a variety of algebras of the same types is discussed. Furthermore, we apply hypersubstitution theory to define a Fix(In,Y)-full closed identity, a Fix(In,Y)-full closed variety and present some concrete examples. [ABSTRACT FROM AUTHOR]

We construct an additive basis for a relatively free alternative algebra of Lie-nilpotent degree 5, describe the associative center and core of this algebra, and find the T-generators of the full center. Also, we give some asymptotic estimate for the codimension of the T-ideal generated by a commutator of degree 5 in a free alternative algebra, and find a finite-dimensional superalgebra that generates the variety of alternative algebras with the Lie-nilpotency of the selfadjoint operator of degree 5. [ABSTRACT FROM AUTHOR]

Join-preserving maps on the discrete time scale ω+, referred to as time warps, have been proposed as graded modalities that can be used to quantify the growth of information in the course of program execution. The set of time warps forms a simple distributive involutive residuated lattice--called the time warp algebra--that is equipped with residual operations relevant to potential applications. In this paper, we show that although the time warp algebra generates a variety that lacks the finite model property, it nevertheless has a decidable equational theory. We also describe an implementation of a procedure for deciding equations in this algebra, written in the OCaml programming language, that makes use of the Z3 theorem prover. [ABSTRACT FROM AUTHOR]

In this paper, we give a basic structure theorem based on the study of extreme cases for the value of ≺ (the classical precedence relation between ultrafilters), i.e., ≺ = ∅ and no isolated element in ≺. This gives rise, respectively, to the temporal varieties O and W , with the result that O generates a variety of temporal algebras. We also characterize the simple temporal algebras by means of arithmetical properties related to basical temporal operators; we conclude that the simplicity of the temporal algebra lies in being able to make 0 any element less than 1 by repeated application to it of the L operator. We then present an algebraic construction similar to a product but in which the temporal operations are not defined componentwise. This new "product" is shown to be useful in the study of algebra order and finding of bounds by means of something similar to a lifting process. Finally, we give an alternative proof of an already known result on atoms counting in free temporal algebras. [ABSTRACT FROM AUTHOR]

To the memory of V. A. Artamonov (1946–2021) All algebras of a certain type are said to form a Nielsen–Schreier variety if every subalgebra of a free algebra is free. This property has been perceived as extremely rare; in particular, only six Nielsen–Schreier varieties of algebras with one binary operation have been discovered in prior work on this topic. We propose an effective combinatorial criterion for the Nielsen–Schreier property in the case of algebras over a field of zero characteristic; in our approach, operads play a crucial role. Using this criterion, we show that the well-known varieties of all pre-Lie algebras and of all Lie-admissible algebras are Nielsen–Schreier, and, quite surprisingly, that there are already infinitely many non-equivalent Nielsen–Schreier varieties of algebras with one binary operation and identities of degree four. [ABSTRACT FROM AUTHOR]

Let be an embedded projective variety. The Lie algebra defined by the tangent directions to at smooth points is an interesting algebraic invariant of . In some cases, this algebra is isomorphic to the symbol algebra of a filtered system of distributions on a Fano manifold, which plays an important role in the theory of these manifolds. In addition, algebras defined by tangent directions are interesting on their own right. In this paper, we study the Lie algebra corresponding to a variety that is the projectivization of the orbit of the lowest weight vector of an irreducible representation of a complex semisimple Lie group. We describe these algebras in terms of generators and relations. In many cases, we can describe their structure completely. [ABSTRACT FROM AUTHOR]

Kleinian singularities, that is, the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory, and Singularity theory. The filtered deformations of these algebras of invariants were classified by Brieskorn (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when the smaller subgroup is normal in the larger. [ABSTRACT FROM AUTHOR]

Let G be a simple algebraic group with ${\mathfrak g}={\textrm{Lie }} G$ and ${\mathcal O}_{\textsf{min}}\subset{\mathfrak g}$ the minimal nilpotent orbit. For a ${\mathbb Z}_2$ -grading ${\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1$ , let $G_0$ be a connected subgroup of G with ${\textrm{Lie }} G_0={\mathfrak g}_0$. We study the $G_0$ -equivariant projections $\varphi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to {\mathfrak g}_0$ and $\psi:\overline{{\mathcal O}_{\textsf{min}}}\to{\mathfrak g}_1$. It is shown that the properties of $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ essentially depend on whether the intersection ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1$ is empty or not. If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then both $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ contain a 1-parameter family of closed $G_0$ -orbits, while if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1=\varnothing$ , then both are $G_0$ -prehomogeneous. We prove that $\overline{G{\cdot}\varphi({\mathcal O}_{\textsf{min}})}=\overline{G{\cdot}\psi({\mathcal O}_{\textsf{min}})}$. Moreover, if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then this common variety is the affine cone over the secant variety of ${\mathbb P}({\mathcal O}_{\textsf{min}})\subset{\mathbb P}({\mathfrak g})$. As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple G -module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of ${\mathfrak g}$ in place of ${\mathfrak g}_0$ or spherical nilpotent G -orbits in place of ${\mathcal O}_{\textsf{min}}$. [ABSTRACT FROM AUTHOR]

We show that free algebras of the variety of algebras generated by the Witt algebra W n , the left-symmetric Witt algebra L n , and the symplectic Poisson algebra P n can be described as subalgebras of differential polynomial algebras with respect to appropriately defined products. Using these representations, we prove that W n , L n , P n , and the free algebras of the varieties of algebras generated by these algebras are equationally Noetherian. [ABSTRACT FROM AUTHOR]

We initiate the study of factorization centers of birational maps and complete it for surfaces over a perfect field in this article. We prove that for every birational automorphism ϕ: X 99K X of a smooth projective surface X over a perfect field k, the blowup centers are isomorphic to the blowdown centers in every weak factorization of ϕ. This implies that nontrivial L-equivalences of zero-dimensional varieties cannot be constructed based on birational automorphisms of a surface. It also implies that rationality centers are well defined for every rational surface X; namely, there exists a zero-dimensional variety intrinsic to X, which is blown up in any rationality construction of X. [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]

We apply, in the context of semigroups, the main theorem from the authors' paper "Algebras defined by equations" (Higgins and Jackson in J Algebra 555:131–156, 2020) that an elementary class C of algebras which is closed under the taking of direct products and homomorphic images is defined by systems of equations. We prove a dual to the Birkhoff theorem in that if the class is also closed under the taking of containing semigroups, some basis of equations of C is free of the ∀ quantifier. We also observe the decidability of the class of equation systems satisfied by semigroups, via a link to systems of rationally constrained equations on free semigroups. Examples are given of EHP-classes for which neither (∀ ⋯) (∃ ⋯) equation systems nor (∃ ⋯) (∀ ⋯) systems suffice. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the structure of irreducible modules in the variety ℳ of commutative power-associative nilalgebras of nilindex ≤ 4. If A ∈ ℳ with dimension at most 5, then we prove that A 2 is contained in the annihilator of every irreducible A -module in the variety ℳ. Also, we consider the enveloping algebra of an algebra A in the variety ℳ and we obtain a new example of a commutative power-associative non-nilpotent nilalgebra of dimension 9. [ABSTRACT FROM AUTHOR]

The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order |$p$| and |$p$| -rings, |$p$| prime and lift this result to the axled Łukasiewicz–Moisil algebra |$L \cong B_s \times P$| and the ring |$\prod ^s F_2 \times \prod ^l F_p$|⁠ , where |$B_s$| is a Boolean algebra of order |$2^s$|⁠ , |$P$| a |$p$| -valued Post algebra of order |$p^l$| and |$F_p$| is the prime field of order |$p$|⁠. [ABSTRACT FROM AUTHOR]

ntly, Gautama algebras were defined and investigated as a common generalization of the variety RDBLSt of regular double Stone algebras and the variety RKLSt of regular Kleene Stone algebras, both of which are, in turn, generalizations of Boolean algebras. Those algebras were named in honor and memory of the two founders of Indian Logic{Akshapada Gautama and Medhatithi Gautama. The purpose of this paper is to define and investigate a generalization of Gautama algebras, called \Almost Gautama algebras (AG, for short)." More precisely, we give an explicit description of subdirectly irreducible Almost Gautama algebras. As consequences, explicit description of the lattice of subvarieties of AG and the equational bases for all its subvarieties are given. It is also shown that the variety AG is a discriminator variety. Next, we consider logicizing AG; but the variety AG lacks an implication operation. We, therefore, introduce another variety of algebras called \Almost Gautama Heyting algebras" (AGH, for short) and show that the variety AGH is term-equivalent to that of AG. Next, a propositional logic, called AG (or AGH), is defined and shown to be algebraizable (in the sense of Blok and Pigozzi) with the variety AG, via AGH; as its equivalent algebraic semantics (up to term equivalence). All axiomatic extensions of the logic AG, corresponding to all the subvarieties of AG are given. They include the axiomatic extensions RDBLSt, RKLSt and G of the logic AG corresponding to the varieties RDBLSt, RKLSt, and G (of Gautama algebras), respectively. It is also deduced that none of the axiomatic extensions of AG has the Disjunction Property. Finally, We revisit the classical logic with strong negation CN and classical Nelson algebras CN introduced by Vakarelov in 1977 and improve his results by showing that CN is algebraizable with CN as its algebraic semantics and that the logics RKLSt, RKLStH, 3-valued Lukasivicz logic and the classical logic with strong negation are all equivalent [ABSTRACT FROM AUTHOR]

By studying the variety of Jónsson–Tarski algebras, we demonstrate two obstacles to the existence of large Jónsson algebras in certain varieties. First, if an algebra J in a language L has cardinality greater than | L | + and a distributive subalgebra lattice, then it must have a proper subalgebra of size |J|. Second, if an algebra J in a language L satisfies cf (| J |) > 2 | L | + and lies in a residually small variety, then it again must have a proper subalgebra of size |J|. We apply the first result to show that Jónsson algebras in the variety of Jónsson–Tarski algebras cannot have cardinality greater than ℵ 1 . We also construct 2 ℵ 1 many pairwise nonisomorphic Jónsson algebras in this variety, thus proving that for some varieties the maximum possible number of Jónsson algebras can be achieved. [ABSTRACT FROM AUTHOR]

This article explores the concepts of pretabularity and interpolation in various logics, including propositional modal and positive logics, as well as J-logics. It discusses the decidable nature of the tabularity problem in intuitionistic logic, minimal logic, and modal logics. The article also examines Craig's interpolation property in different logics, providing descriptions and decidable results for superintuitionistic, positive, and modal logics. The text utilizes algebraic semantics to establish correspondences between logics and algebraic structures. While the tabularity and pretabularity problems are decidable in certain logics, there are still open questions regarding the interpolation problem in the class of extensions of minimal logic J and modal logic K. The decidability of the interpolation property over J and the exact number of extensions of S4 with the interpolation property remain unknown. [Extracted from the article]

It is introduced a new algebra (A , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1) called L P G -algebra if (A , ⊗ , ⊕ , ∗ , 0 , 1) is L P -algebra (i.e. an algebra from the variety generated by perfect MV-algebras) and (A , ⇀ , 0 , 1) is a Gödel algebra (i.e. Heyting algebra satisfying the identity (x ⇀ y) ∨ (y ⇀ x) = 1) . The lattice of congruences of an L P G -algebra (A , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1) is isomorphic to the lattice of Skolem filters (i.e. special type of MV-filters) of the MV-algebra (A , ⊗ , ⊕ , ∗ , 0 , 1) . The variety L P G of L P G -algebras is generated by the algebras (C , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1) where (C , ⊗ , ⊕ , ∗ , 0 , 1) is Chang MV-algebra. Any L P G -algebra is bi-Heyting algebra. The set of theorems of the logic L P G is recursively enumerable. Moreover, we describe finitely generated free L P G -algebras. [ABSTRACT FROM AUTHOR]

We prove the Fargues--Rapoport conjecture for p-adic period domains in the non-basic case with minuscule cocharacter. More precisely, we give a group-theoretical criterion for the cases when the admissible locus and weakly admissible locus coincide. This generalizes the result of Hartl (2013) for the group GLn. In the last section, we also give a conjecture about the intersection of weakly admissible locus and the Newton strata in the flag variety. [ABSTRACT FROM AUTHOR]

We characterise the projective varieties related to the second row of the Freudenthal-Tits magic square, for both the split and the non-split form, using a common, simple and short geometric axiom system. A special case of our result simultaneously captures the analogues over arbitrary fields of the Severi varieties (comprising the 27-dimensional E6 module and some of its subvarieties), as well as the Veronese representations of projective planes over composition division algebras (most notably the Cayley plane). It is the culmination of almost four decades of work since the original 1984 result by Mazzocca and Melone who characterised the quadric Veronese variety over a finite field of odd order. The latter result is a finite counterpart to the characterisation of the complex quadric Veronese surface by Severi from 1901. [ABSTRACT FROM AUTHOR]

Symmetric and k-cyclic structure of modal pseudocomplemented De Morgan algebras was introduced previously. In this paper, we first present the construction of epimorphisms between finite symmetric (or 2-cyclic) modal pseudocomplemented De Morgan algebras. Furthermore, we compute the cardinality of the set of all epimorphism between finite structures. Secondly, we present the construction of finite free algebras on the variety of k-cyclic modal pseudocomplemented De Morgan algebras and display how our computations are in fact generalizations to others in the literature. Our work is strongly based on the properties of epimorphisms and automorphisms and the fact that the variety is finitely generated. [ABSTRACT FROM AUTHOR]

This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means Hilbert-style calculi that are finite (i.e. that contain a finite number of finitary rule schemata). On the negative side, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely axiomatizable; we likewise establish that the order-preserving logic of the variety of all Ockham algebras is also not finitely axiomatizable. On the positive side, we show that an arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite number of equations if and only if the corresponding order-preserving logic is axiomatized by a finite Hilbert-style calculus. This equivalence also holds for every subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a new proof that the implication-free fragment of intuitionistic logic is finitely axiomatizable, as well as a new corresponding Hilbert-style calculus. Our proofs are constructive in that they allow us to effectively convert an equational presentation of a variety of algebras into a Hilbert-style calculus for the corresponding order-preserving logic, and vice versa. We also consider the assertional logics associated to the above-mentioned varieties, showing in particular that the assertional logics of finitely axiomatizable subvarieties of semi-De Morgan algebras are finitely axiomatizable as well. [ABSTRACT FROM AUTHOR]

We use the left self‐distributive axiom to introduce and study a special class of weak Heyting algebras, called self‐distributive weak Heyting algebras (SDWH‐algebras). We present some useful properties of SDWH‐algebras and obtain some equivalent conditions of them. A characteristic of SDWH‐algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH‐algebras and some of the known subvarieties of weak Heyting algebras such as the variety of Heyting algebras, the variety of basic algebras, the variety of subresiduated lattices, the variety of reflexive WH‐algebras (RWH‐algebras), and the variety of transitive WH‐algebras (TWH‐algebras). [ABSTRACT FROM AUTHOR]

A subresiduated lattice is a pair (A,D)$(A,D)$, where A is a bounded distributive lattice, D is a bounded sublattice of A and for every a,b∈A$a,b\in A$ there exists the maximum of the set {d∈D:a∧d≤b}$\lbrace d\in D:a\wedge d\le b\rbrace$, which is denoted by a→b$a\rightarrow b$. This pair can be regarded as an algebra (A,∧,∨,→,0,1)$(A,\wedge ,\vee ,\rightarrow ,0,1)$ of type (2, 2, 2, 0, 0), where D={a∈A:1→a=a}$D=\lbrace a\in A: 1\rightarrow a =a\rbrace$. The class of subresiduated lattices is a variety which properly contains the variety of Heyting algebras. In this paper we study the subvariety of subresiduated lattices, denoted by S□$\mathrm{S}^{\Box }$, whose members satisfy the equation 1→(a∨b)=(1→a)∨(1→b)$1\rightarrow (a\vee b) = (1\rightarrow a) \vee (1\rightarrow b)$. Inspired by the fact that in any subresiduated lattice whose order is total the previous equation and the condition a→b∈{1→b,1}$a\rightarrow b \in \lbrace 1\rightarrow b,1\rbrace$ for every a,b$a,b$ are satisfied, we also study the subvariety of S□$\mathrm{S}^{\Box }$ generated by the class whose members satisfy that a→b∈{1→b,1}$a\rightarrow b \in \lbrace 1\rightarrow b,1\rbrace$ for every a,b$a,b$. [ABSTRACT FROM AUTHOR]

A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere's classical characterization of varieties of ordinary algebras. We also study the relationship of varieties to discrete Lawvere theories, and varieties as concrete categories over Pos . [ABSTRACT FROM AUTHOR]

The Conference on Theoretical and Computational Algebra 2023 (TCA 2023) took place in Pocinho, Portugal, attracting over 80 researchers from Europe, Australia, Brazil, Canada, and the USA. The conference covered various topics in algebra and automated reasoning, with a significant focus on semigroup theory. As a result, the editors of 'Semigroup Forum' decided to dedicate a special issue to selected papers from TCA 2023. This issue includes 11 papers that underwent a rigorous refereeing process, with additional papers still under review for future publication. The conference also honored Dr. Gracinda M. S. Gomes, a retired professor from the University of Lisbon, who made significant contributions to semigroup theory and fostered collaboration between Portugal and the UK. [Extracted from the article]

The "semilattices of Mal'cev blocks", for short SMB algebras, were defined by A. Bulatov. In a recently accepted paper by P. Đapić, P. Marković, R. McKenzie, and A. Prokić, the class of all SMB algebras and its subclass of regular SMB algebras were proved to be varieties of algebras. In this paper, we find an equational base of the first variety and simplify the previously known equational base of the second variety. [ABSTRACT FROM AUTHOR]

We introduce Boolean-like algebras of dimension n ( n BA s) having n constants e 1 , ... , e n , and an (n + 1) -ary operation q (a "generalised if-then-else") that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of n BA s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The n BA s provide the algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, n CL , and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in n CL , and, via the embeddings, in all the finite tabular logics. [ABSTRACT FROM AUTHOR]

A point p in a projective space is h-identifiable via a variety X if there is a unique way to write p as a linear combination of h points of X. Identifiability is important both in algebraic geometry and in applications. In this paper we propose an entirely new approach to study identifiability, connecting it to the notion of secant defect for any smooth projective variety. In this way we are able to improve the known bounds on identifiability and produce new identifiability statements. In particular, we give optimal bounds for some Segre and Segre-Veronese varieties and provide the first identifiability statements for Grassmann varieties. [ABSTRACT FROM AUTHOR]