Alpha Ideals and the Space of Prime Alpha Ideals in Universal Algebras

The purpose of this paper is to study α -ideals in a more general context, in universal algebras having a constant 0 . Several characterizations are obtained for an ideal I of an algebra A to be an α -ideal. It is shown that the class of all α -ideals of an algebra A forms an algebraic lattice. Prime α -ideals and several related properties are investigated. Some properties of the spectral space of prime α -ideals equipped with the hull-kernel topology are derived.

L-fuzzy cosets in universal algebras

In this paper, we introduce the notion of fuzzy costs in a more general context, in universal algebra by the use of coset terms. We study the structure of fuzzy cosets by applying the general theory of algebraic fuzzy systems. Fuzzy cosets generated by a fuzzy set are characterized in different ways. It is also proved that the class of fuzzy cosets determined by an element forms an algebraic closure fuzzy set system. Finally, we give a set of necessary and sufficient conditions for a given variety of algebras to be congruence permutable by applying the theory of fuzzy cosets.

Pseudo-free families of computational universal algebras

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 $\eta (|d|)\left( \text{ resp}\text{., }\left| {{H}_{d}} \right|\le {{2}^{\eta (|d|)}} \right).$ 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 = Ω0 ∪ {ω}, where Ω0 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.

Admissible Galois Structures on the categories dual to some varieties of universal algebras

The subject of the paper is suggested by G. Janelidze and motivated by his earlier result giving a positive answer to the question posed by S. MacLane whether the Galois theory of homogeneous linear ordinary differential equations over a differential field (which is Kolchin–Ritt theory and an algebraic version of Picard–Vessiot theory) can be obtained as a particular case of G. Janelidze’s Galois theory in categories. One ground category in the Galois structure involved in this theory is dual to the category of commutative rings with unit, and another one is dual to the category of commutative differential rings with unit. In the present paper, we apply the general categorical construction, the particular case of which gives this Galois structure, by replacing “commutative rings with unit” by algebras from any variety \mathscr{V} of universal algebras satisfying the amalgamation property and a certain condition (of the syntactical nature) for elements of amalgamated free products which was introduced earlier, and replacing “commutative differential rings with unit” by \mathscr{V}-algebras equipped with additional unary operations which satisfy some special identities to construct a new Galois structure. It is proved that this Galois structure is admissible. Moreover, normal extensions with respect to it are characterized in the case where \mathscr{V} is any of the following varieties: abelian groups, loops and quasigroups.

Annihilators in Universal Algebras: A New Approach

The purpose of this paper is to study annihilators and annihilator ideals in a more general context; in universal algebras.