Nullary Operations Research Articles

Exploring Pointed Sets and their Role in the Category of Pointed F-Sets: A Study on Kernels in the Category F-Sets

This paper delves into the realm of pointed sets and their significance within the framework of algebraic structures, particularly focusing on their role in the category of pointed F-sets. Pointed sets, characterized by a single nullary operation identifying a base point, serve as fundamental components in algebraic reasoning, with applications in various mathematical domains. The category of pointed F-sets, comprising pointed sets as objects and pointed F-associations as morphisms, offers a rich ground for exploring the behavior of specialized functions. Central to our investigation is the demonstration of kernels within the category F-sets, elucidating the foundational properties and structural insights underlying pointed sets and their associations.

A new model of the free monogenic digroup

It is well-known that one of open problems in the theory of Leibniz algebras is to find asuitable generalization of Lie’s third theorem which associates a (local) Lie group to any Liealgebra, real or complex. It turns out, this is related to finding an appropriate analogue of a Liegroup for Leibniz algebras. Using the notion of a digroup, Kinyon obtained a partial solution ofthis problem, namely, an analogue of Lie’s third theorem for the class of so-called split Leibnizalgebras. A digroup is a nonempty set equipped with two binary associative operations, aunary operation and a nullary operation satisfying additional axioms relating these operations.Digroups generalize groups and have close relationships with the dimonoids and dialgebras,the trioids and trialgebras, and other structures. Recently, G. Zhang and Y. Chen applied themethod of Grobner–Shirshov bases for dialgebras to construct the free digroup of an arbitraryrank, in particular, they considered a monogenic case separately. In this paper, we give a simplerand more convenient digroup model of the free monogenic digroup. We construct a new classof digroups which are based on commutative groups and show how the free monogenic groupcan be obtained from the free monogenic digroup by a suitable factorization.

Partial Menger algebras and their weakly isomorphic representation

As generalization of semigroups, Karl Menger introduced in the 1940th algebras of multiplace operations. Such algebras satisfy the superassociative law, a generalization of the associative law. Menger algebras are defined as models of this superassociative law. Cayley’s theorem for semigroups says that any model of the associative law is isomorphic to a transformation semigroup. R. M. Dicker proved in 1963 that every Menger algebra of rank [Formula: see text] is isomorphic to a Menger algebra of [Formula: see text]-ary operations on some set. The composition of terms in which each variable occurs at most [Formula: see text]-times, so-called [Formula: see text]-terms, leads to partial algebras where the superassociative law is satisfied as a weak identity. In this paper, we introduce the concepts of a partial Menger algebra, a unitary partial Menger algebra and of a generalized partial Menger algebra. We prove that [Formula: see text]-terms of some type form a generalized partial Menger algebra with infinitely many nullary operations. Using weak identities and weak isomorphisms, Dicker’s result will be extended to partial Menger algebras and to unitary partial Menger algebras.

Pseudo-free families of computational universal algebras

Abstract Let Ω be a finite set of finitary operation symbols. We initiate the study of (weakly) pseudo-free families of computational Ω-algebras in arbitrary varieties of Ω-algebras. A family (Hd | d ∈ D) of computational Ω-algebras (where D ⊆ {0, 1}*) is called polynomially bounded (resp., having exponential size) if there exists a polynomial η such that for all d ∈ D, the length of any representation of every h ∈ Hd is at most η ( | d | ) ( resp ., | H d | ≤ 2 η ( | d | ) ) . $\eta (|d|)\left( \text{ resp}\text{., }\left| {{H}_{d}} \right|\le {{2}^{\eta (|d|)}} \right).$ First, we prove the following trichotomy: (i) if Ω consists of nullary operation symbols only, then there exists a polynomially bounded pseudo-free family; (ii) if Ω = Ω 0 ∪ {ω}, where Ω 0 consists of nullary operation symbols and the arity of ω is 1, then there exist an exponential-size pseudo-free family and a polynomially bounded weakly pseudo-free family; (iii) in all other cases, the existence of polynomially bounded weakly pseudo-free families implies the existence of collision-resistant families of hash functions. In this trichotomy, (weak) pseudo-freeness is meant in the variety of all Ω-algebras. Second, assuming the existence of collision-resistant families of hash functions, we construct a polynomially bounded weakly pseudo-free family and an exponential-size pseudo-free family in the variety of all m-ary groupoids, where m is an arbitrary positive integer.

Partial clones

A set [Formula: see text] of operations defined on a nonempty set [Formula: see text] is said to be a clone if [Formula: see text] is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the [Formula: see text]-ary operations defined on set [Formula: see text] for all natural numbers [Formula: see text] and the operations are the so-called superposition operations [Formula: see text] for natural numbers [Formula: see text] and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set [Formula: see text] and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.

Semilattice sums of algebras and Mal’tsev products of varieties

The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider the question of when this quasivariety is a variety. The main result asserts that if $$\mathcal {V}$$ is a strongly irregular variety with no nullary operations and at least one non-unary operation, and $$\mathcal {S}$$ is the variety, of the same type as $$\mathcal {V}$$ , equivalent to the variety of semilattices, then the Mal’tsev product $$\mathcal {V}\circ \mathcal {S}$$ is a variety. It consists precisely of semilattice sums of algebras in $$\mathcal {V}$$ . We derive an equational base for the product from an equational base for $$\mathcal {V}$$ . However, if $$\mathcal {V}$$ is a regular variety, then the Mal’tsev product may not be a variety. We discuss various applications of the main result, and examine some detailed representations of algebras in $$\mathcal {V}\circ \mathcal {S}$$ .

On the ascending and descending chain conditions in the lattice of monoid varieties

In this work we consider monoids as algebras with an associative binary operation and the nullary operation that fixes the identity element. We found an example of two varieties of monoids with finite subvariety lattices such that their join covers one of them and has a continuum cardinality subvariety lattice that violates the ascending chain condition and the descending chain condition.

The Lattice of Varieties of Implication Semigroups

In 2012, the second author introduced and examined a new type of algebras as a generalization of De Morgan algebras. These algebras are of type (2,0) with one binary and one nullary operation satisfying two certain specific identities. Such algebras are called implication zroupoids. They invesigated in a number of articles by the second author and J.M.Cornejo. In these articles several varieties of implication zroupoids satisfying the associative law appeared. Implication zroupoids satisfying the associative law are called implication semigroups. Here we completely describe the lattice of all varieties of implication semigroups. It turns out that this lattice is non-modular and consists of 16 elements.

Algebraization of Inferring Functional Dependences in Relational Databases

The paper considers a closed automaton with six not more than two-address instructions called inference axioms. Its purpose is to transform the so-called functional dependences over an arbitrary finite set R called a scheme. We establish the possibility of a functionally equivalent replacement of algorithms for inferring functional dependences by algebraic expressions with one of the binary algebraic operations ▶ and ▷ with the addition of very simple nullary and unary operations (instructions) to them, which together comprise the universal D-algebra.

О КОНГРУЭНЦ–КОГЕРЕНТНЫХ АЛГЕБРАХ РИСА И АЛГЕБРАХ С ОПЕРАТОРОМ

The paper contains a classification of congruence-coherent Rees algebras and algebras with an operator. The concept of coherence was introduced by D.Geiger. An algebra A is called coherent if each of its subalgebras containing a class of some congruence on A is a union of such classes. In Section 3 conditions for the absence of congruence-coherence property for algebras having proper subalgebras are found. Necessary condition of congruence-coherence for Rees algebras are obtained. Sufficient condition of congruence-coherence for algebras with an operator are obtained. In this section we give a complete classification of congruence-coherent unars. In Section 4 some modification of the congruence-coherent is considered. The concept of weak and locally coherence was introduced by I.Chajda. An algebra A with a nullary operation 0 is called weakly coherent if each of its subalgebras including the kernel of some congruence on A is a union of classes of this congruence. An algebra A with a nullary operation 0 is called locally coherent if each of its subalgebras including a class of some congruence on A also includes a class the kernel of this congruence. Section 4 is devoted to proving sufficient conditions for algebras with an operator being weakly and locally coherent. In Section 5 deals with algebras ⟨ A,d,f ⟩ with one ternary operation d ( x,y,z ) and one unary operation f acting as endomorphism with respect to the operation d ( x,y,z ). Ternary operation d ( x,y,z ) was defined according to the approach offered by V.K. Kartashov. Necessary and sufficient conditions of congruence-coherent for algebras ⟨ A,d,f ⟩ are obtained. Also, necessary and sufficient conditions of weakly and locally coherent for algebras ⟨ A,d,f, 0⟩ with nullary operation 0 for which f (0) = 0 are obtained.

SUPER-BOOLEAN FUNCTIONS AND FREE BOOLEAN QUASILATTICES

A Boolean quasilattice is an algebra with hyperidentities of the variety of Boolean algebras. In this paper, we give a functional representation of the free n-generated Boolean quasilattice with two binary, one unary and two nullary operations. Namely, we define the concept of super-Boolean function and prove that the free Boolean quasilattice with two binary, one unary and two nullary operations on n free generators is isomorphic to the Boolean quasilattice of super-Boolean functions of n variables.

Clones with Nullary Operations

This article discusses clones with nullary operations and the corresponding relational clones, both defined on arbitrary sets. By means of two pairs of kernel and closure operators, the relationship between such clones and clones in the traditional sense, i.e. without nullary operations, is investigated, and in particular the latter type of clones is located in the lattice of all clones. Finally, the fundamental theorem characterising Galois closed sets w.r.t. an adjusted version of Pol – Inv as local closures of clones and relational clones, respectively, is proven in the more comprehensive setting.

Category Theoretic Understandings of Universal Algebra and its Dual: Monads and Lawvere Theories, Comonads and What?

Universal algebra is often known within computer science in the guise of algebraic specification or equational logic. In 1963, it was given a category theoretic characterisation in terms of what are now called Lawvere theories. Unlike operations and equations, a Lawvere theory is uniquely determined by its category of models. Except for a caveat about nullary operations, the notion of Lawvere theory is equivalent to the universal algebraistʼs notion of an abstract clone. Lawvere theories were soon followed by a further characterisation of universal algebra in terms of monads, the latter quickly becoming preferred by category theorists but not by universal algebraists. In the 1990ʼs began a systematic attempt to dualise the situation. The notion of monad dualises to that of comonad, providing a framework for studying transition systems in particular. Constructs in universal algebra have begun to be dualised too, with different leading examples. But there is not yet a definitive dual of the concept of Lawvere theory, or that of abstract clone, or even a definitive dual of operations and equations. We explore the situation here.

Relative universality and universality obtained by adding constants

A variety \({\mathbb{V}}\) is var-relatively universal if it contains a subvariety \({\mathbb{W}}\) such that the class of all homomorphisms that do not factorize through any algebra in \({\mathbb{W}}\) is algebraically universal. And \({\mathbb{V}}\) has an algebraically universal α-expansion \({\alpha\mathbb{V}}\) if adding α nullary operations to all algebras in \({\mathbb{V}}\) gives rise to a class \({\alpha\mathbb{V}}\) of algebras that is algebraically universal. The first two authors have conjectured that any varrelative universal variety \({\mathbb{V}}\) has an algebraically universal α-expansion \({\alpha\mathbb{V}}\) . This note contains a more general result that proves this conjecture.

On congruence distributivity of ordered algebras with constants

We define the order-congruence distributivity at 0 and order- congruence n-distributivity at 0 of ordered algebras with a nullary op- eration 0. These notions are generalizations of congruence distributiv- ity and congruence n-distributivity. We prove that a class of ordered algebras with a nullary operation 0 closed under taking subalgebras and direct products is order-congruence distributive at 0 iff it is order- congruence n-distributive at 0. We also characterize such classes by a Mal'tsev condition.

Embedding general algebras into modules

The problem of embedding general algebras into modules is revisited. We provide a new method of embedding, based on Ježek's embedding into semimodules. We obtain several interesting consequences: a simpler syntactic characterization of algebras, a proof that algebras without nullary operations are actually quasi-linear, and several facts regarding the abelian iff quasi-affine problem.

Mapping Extension of an Algebra

A new construction of algebras called a mapping extension of an algebra is here introduced. The construction yields a generalization of some classical constructions such as the nilpotent extension of an algebra, inflation of a semigroup but also the square extension construction introduced recently for idempotent groupoids. The mapping extension construction is defined for algebras of any fixed type, however nullary operation symbols are here not admitted. It is based on the notion of a retraction and some system of mappings. A mapping extension of a given algebra is constructed as a counterimage algebra by a specially defined retraction. Varieties of algebras satisfying an identity φ(x) ≈ x for a term φ not being a variable (such as varieties of lattices, Boolean algebras, groups and rings) are especially interesting because for such a variety [Formula: see text], all mapping extensions by φ of algebras from [Formula: see text] form an equational class. In the last section, combinatorial properties of the mapping extension construction are considered.

Double Basic Algebras

To any bounded lattice with section antitone bijections can be assigned an algebra with two binary, two unary and a nullary operations satisfying similar axioms as a basic algebra. Due to the doubled similarity type, this algebra is called a double basic algebra. Also conversely, every double basic algebra induces a bounded lattice equipped with antitone bijections in every section. We study properties of double basic algebras, their interval algebras and several conclusions of the so-called pseudo-commutativity.

Subvarieties of the clone extension of the variety of distributive lattices

Let τ be a type of algebras without nullary fundamental operation symbols. An identity . φ ≈ ψ of type τ we call clone compatible if φ and ψ are the same variable or the sets of fundamental operation symbols in φ and ψ are non empty and identical. For a variety ν of type τ we denote by ν c the variety of type τ defined by all clone compatible identities from Id(ν). We call ν c the clone extension of ν. In this paper we describe identities, equational bases and minimal generics of all subvarieties of \( {\user1{{\mathcal{D}}}}^{c} \), where $$ {\user1{{\mathcal{D}}}} $$ is the variety of distributive lattices.

Clones of implicit operations

There is a close connection between a variety and its clone. The clone of a variety is a multibased algebra, where the different universes are the sets of n-ary terms over this variety for every natural number n and where the operations describe the superposition of terms of different arities. All projections are added as nullary operations. Subvarieties correspond to homomorphic images of clones. Subclones can be described by reducts of varieties, isomorphic clones by equivalent varieties. Clone identities correspond to hyperidentities and varieties of clones to hypervarieties. Pseudovarieties are classes of finite algebras which are closed under taking of subalgebras, homomorphic images and finite direct products. Pseudovarieties are important in the theories of finite state automata, rational languages, finite semigroups and their connections. In a very natural way, there arises the question for the clone of a pseudovariety. In the present paper, we will describe this algebraic structure.