A characterization theorem for weak varieties

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let U f denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some U f -algebra. For any U-algebraAsetU A =U i eI({i}×(A K i—domf i A )), where (K i) i eI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, U f ) be similarly defined). Then for every nonempty setM there exists a setN satisfyingM ⊆N such that there exists a bijective mapping σ:U F(N, U) →N −M satisfying σ((i, α)) ∉ α(K i ) for all (i, α) ∈U F (N, U), and, for the structureg=(g i)ieI defined by ,g i : =f i F(N, U) ∪ {(α, σ((i, α))) | (i, α ∈U F(N, U)} id M induces an isomorphism betweenF(M, U f ), and (F(N, U)g).

Some characterizations of partial algebras

This is a paper on Stone duality in computer science with special focus on topics with applications in formal language theory. In Section 2 we give a general overview of Stone duality in its various forms: for Boolean algebras, distributive lattices, and frames. For distributive lattices, we discuss both Stone and Priestley duality. We identify how to move between the different dualities and which dual spaces carry the Scott topology. We then focus on three themes. The first theme is additional operations on distributive lattices and Boolean algebras. Additional operations arise in denotational semantics in the form of predicate transformers. In verification they occur in the form of modal operators. They play an essential role in Eilenberg's variety theorem in the form of quotient operations. Quotient operations are unary instantiations of residual operators which are dual to the operations in the profinite algebras of algebraic language theory. We discuss additional operations in Section 3. The second theme is that of hyperspaces, that is, spaces of subsets of an underlying space. Some classes of algebras may be seen as the class of algebras for a functor. In the case of predicate transformers the dual functors are hyperspace constructions such as the Plotkin, Smyth, and Hoare powerdomain constructions. The algebras-for-a-functor point of view is central to the coalgebraic study of modal logic and to the solution of domain In the algebraic theory of formal languages various hyperspace-related product constructions, such as block and Schutzenberger products, are used to study complexity hierarchies. We describe a construction, similar to the Schutzenberger product, which is dual to adding a layer of quantification to formulas describing formal languages. We discuss hyperspaces in Section 4. The final theme is that of equations. These are pairs of elements of dual spaces. They arise via the duality between subalgebras and quotient spaces and have provided one of the most successful tools for obtaining decidability results for classes of regular languages. The perspective provided by duality allows us to obtain a notion of equations for the study of arbitrary formal languages. Equations in language theory is the topic of Section 5.

A functorial semantics for multi-algebras and partial algebras, with applications to syntax

An investigation of two-dimensional exactly and completely integrable dynamical systems associated with the local part of an arbitrary Lie algebra\(\mathfrak{g}\) whose grading is consistent with an arbitrary integral embedding of 3d-subalgebra in\(\mathfrak{g}\) has been carried out. We have constructed in an explicit form the corresponding systems of nonlinear partial differential equations of the second order and obtained their general solutions in the sense of a Goursat problem. A method for the construction of a wide class of infinite-dimensional Lie algebras of finite growth has been proposed.

Varieties of algebras

An algebraic approach to the theory of regular languages of finite and infinite words (∞-languages) is presented. It extends the algebraic theory of regular languages of finite words, which is based on finite semigroups. Their role is taken over by a structure called right binoid. A variety theorem is proved: there is a one-to-one correspondence between varieties of ∞-languages and pseudovarieties of right binoids. The class of locally threshold testable languages and several natural subclasses (such as the class of locally testable languages) as well as classes of the Borel hierarchy over the Cantor space (restricted to regular languages) are investigated as examples for varieties of ∞-languages. The corresponding pseudovarieties of right binoids are characterized and in some cases defining equations are derived. The connections with the algebraic description and classification of regular languages of infinite words in terms of finite semigroups are pointed out.

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman’s theorem can be derived from it.

In this paper, we give a characterization theorem for rough sets based on quasi order. We obtain an algebra on the rough sets system determined by a quasi order which is the generalization of the algebra of rough sets system determined by an equivalence relation given in [1]. The properties of this algebra are abstracted at various levels and define new class of algebras. Further we give a representation theorem for the new class of algebras.

Over fields of prime characteristic the centre of the universal enveloping algebra of a finite-dimensional Lie algebra contains a finitely generated polynomial algebra over which the universal envelope is a finitely generated module.This result, which is due to Curtis, is crucial in certain investigations of finitely generated soluble Lie algebras and motivates the introduction of the class Max-cu, which will be called the class of Curtis algebras, consisting of Lie algebras whose universal envelopes have the property described above.It has been an open question whether the class Max-cu consists of finite-dimensional Lie algebras.This paper gives an affirmative answer to the question.1. Notation and preliminary results.We employ the notation and conventions of Amayo and Stewart [1].If L is a Lie algebra then U(L) denotes its universal enveloping algebra.We denote by F the class of finite-dimensional Lie algebras;L G Max if every subalgebra is finitely generated;L G Max-u if U(L) is right noetherian; L G Max-cu if there exists a finitely generated polynomial subring R of the centre of U(L) and a finite number of elements "...," in U(L) such that U(L) = Rux + + Run (equivalently U(L) is a finitely generated P-module).A denotes the class of abelian Lie algebras.By the results of Amayo and Stewart [2], we have Max-cu < Max-u < Max, and F < Max-u < Max.Over fields of characteristic zero, F n A < Max-cu and over fields of prime characteristic F < Max-cu.Let L be a Lie algebra defined over a field k, K an extension field of k, U(L) the universal envelope of L and L0 = L k K, the K-Lie algebra extension of L. Then U(L0) = U(L) k K. Thus if L G Max-cu and R = k[z" . . ., z J centre of U(L) such that U(L) = Rux + + Ru", then U(L0) = P0w, + + RQun, where P0 = K[z,,. .., zn], whence L0 G Max-cu as a Lie algebra over K.We note also that the class Max-cu is closed under homomorphic images and finite direct sums.Lie algebras in the class Max-cu will be called Curtis algebras.The proof of the following lemma may be found in Curtis [4] and Amayo and Stewart [1,.Lemma 1.1.Every irreducible module of a Curtis algebra is finite-dimensional.

Varieties of nilpotent Lie algebras of dimension less than six

SC α, CA α, QA α and QEA α stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension α, respectively. Generalizing a result of Nemeti on cylindric algebras, we show that for K ∈ {SC, CA, QA, QEA} and ordinals α < β, the class Nr α K β of α-dimensional neat reducts of β-dimensional K algebras, though closed under taking homomorphic images and products, is not closed under forming subalgebras (i.e. is not a variety) if and only if α > 1. From this it easily follows that for 1 < α < β, the operation of forming α-neat reducts of algebras in K β does not commute with forming subalgebras, a notion to be made precise. We give a contrasting result concerning Halmos' polyadic algebras (with and without equality). For such algebras, we show that the class of infinite dimensional neat reducts forms a variety. We comment on the status of the property of neat reducts commuting with forming subalgebras for various reducts of polyadic algebras that are also expansions of cylindric-like algebras. We try to draw a borderline between reducts that have this property and reducts that do not. Following research initiated by Pigozzi, we also emphasize the strong tie that links the (apparently non-related) property of neat reducts commuting with forming subalgebras with proving amalgamation results in cylindric-like algebras of relations. We show that, like amalgamation, neat reducts commuting with forming subalgebras is another algebraic expression of definability and, accordingly, is also strongly related to the well-known metalogical properties of Craig, Beth and Robinson in the corresponding logics.

A class of finite monounary algebras is said to be a formation if it is closed under homomorphic images and finite subdirect products. Let us denote by \({\mathbb{F}}\) the collection of all formations of finite monounary algebras. Then \({\mathbb{F}}\) can be partially ordered by the class-theoretical inclusion and it forms a complete lattice. The main aim of this paper is to show that \({\mathbb{F}}\) is isomorphic to the lattice of all hereditary subsets of a certain poset.

The notion of a Boolean algebra with operators (BAO) was first defined by Jonsson and Tarski in 1951. Since that time, many varieties of BAOs, including modal algebras, closure algebras, monadic algebras, and of course relation algebras, have been studied. With the exception of relation algebras, these varieties each have one unary operator. This paper investigates a variety of BAOs with one binary operator. Begin with a semigroup S = $(S,\cdot).$ The complex algebra of S, denoted S$\sp+,$ is a Boolean algebra whose underlying set is the power set of S with set union, intersection and complementation as the Boolean algebra operations. The multiplication operation defined on the semigroup induces a normal, associative binary operation, '*', on the complex algebra as follows: for all subsets A and B contained in S, $A\* B=\{a\cdot b:a\in A, b\in B\}.$ Hence, the complex algebra of a semigroup is a BAO with one normal binary associative operator. Let S$\sp+$ be the class of all complex algebras of semigroups and consider the variety generated by the class S$\sp+,$ denoted ${\cal V}$(S$\sp+).$ The tools required to study this variety are developed, including the duality between BAOs and relational structures as it applies to ${\cal V}$(S$\sp+).$ A closure operator is defined which is used to determine homomorphic images of members of ${\cal V}$(S$\sp+).$ Theorems on the subdirectly irreducible and simples algebras in ${\cal V}$(S$\sp+)$ are proved. Next, the structure of ${\cal V}$(S$\sp+)$ is analyzed. The general problem of representing a BAO with one binary, normal, associative operator as a member of ${\cal V}$(S$\sp+)$ is discussed. Several examples and theorems concerning representation are presented. It is shown that the quasivariety generated by S$\sp+$ is strictly contained in ${\cal V}$(S$\sp+).$ Lastly, the structure of the lattice of subvarieties of ${\cal V}$(S$\sp+)$ is investigated. There are precisely two atoms in this lattice; each atom is generated by a two element algebra. Two infinite chains of varieties exist, with the smallest element in each chain a cover of exactly one atom. This leads to a discussion of certain splitting algebras and conjugate varieties. Finally, equations characterizing some important subvarieties are developed.

The goal of this paper is to generalize a notion of characteristic (or Jankov) formula by using finite partial Heyting algebras instead of the finite subdirectly irreducible algebras: with every finite partial Heyting algebra we associate a characteristic formula, and we study the properties of these formulas. We prove that any intermediate logic can be axiomatized by such formulas. We further discuss the correlations between characteristic formulas of finite partial algebras and canonical formulas. Then with every well-connected Heyting algebra we associate a set of characteristic formulas that correspond to each finite relative subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular characteristic formulas. In the last section we outline an approach how to generalize these obtained results to the broad classes of algebras.