Commutative Groupoids Research Articles

Об эндоморфизмах аддитивного моноида подсетей двухслойной нейронной сети

Previously, for each multilayer neural network of direct signal propagation (hereinafter, simply a neural network), finite commutative groupoids were introduced, which were called additive subnet groupoids. These groupoids are closely related to the subnets of the neural network over which they are built. A grupoid is a monoid if and only if it is built over a two-layer neural network. Earlier, endomorphisms and their properties were studied for these groupoids. Some endomorphisms were constructed, but an exhaustive element-by-element description was not received. It was shown that every finite monoid is isomorphic to some submonoid of the monoid of all endomorphisms of a suitable additive subnet groupoid for some suitable neural network. In this paper, we study endomorphisms of additive groupoids of subnets of twolayer neural networks. The main result of the work is an element-wise description of the monoid of all endomorphisms of additive monoids of subnets built over a two-layer neural network. The item-by-item description is obtained by constructing a general form of endomorphism. The general view of an endomorphism is parameterized by the endomorphisms of suitable booleans with respect to the union operation. Therefore, endomorphisms of these Booleans were studied in this work. In particular, the semirings of endomorphisms of these Booleans with respect to the union were studied. In addition, to describe the general form of the endomorphism of the additive monoid of subnets, homomorphisms of one Boalean into another (with respect to union) were used.

Sheffer operation in relational systems

The concept of a Sheffer operation known for Boolean algebras and orthomodular lattices is extended to arbitrary directed relational systems with involution. It is proved that to every such relational system, there can be assigned a Sheffer groupoid and also, conversely, every Sheffer groupoid induces a directed relational system with involution. Hence, investigations of these relational systems can be transformed to the treatment of special groupoids which form a variety of algebras. If the Sheffer operation is also commutative, then the induced binary relation is antisymmetric. Moreover, commutative Sheffer groupoids form a congruence distributive variety. We characterize symmetry, antisymmetry and transitivity of binary relations by identities and quasi-identities satisfied by an assigned Sheffer operation. The concepts of twist products of relational systems and of Kleene relational systems are introduced. We prove that every directed relational system can be embedded into a directed relational system with involution via the twist product construction. If the relation in question is even transitive, then the directed relational system can be embedded into a Kleene relational system. Any Sheffer operation assigned to a directed relational system {mathbf {A}} with involution induces a Sheffer operation assigned to the twist product of {mathbf {A}}.

On block commutative groupoids

In this paper, we introduce the notion of a block commutativity in several groupoids, and show that the class of block commutative groupoids and the class of d/BCK-algebras are Smarandache disjoint. The block commutativity in linear/quadratic groupoids is investigated, and we prove that every group is a normal groupoid. Moreover, we discuss block n-commutative groupoids and block ranks.

Non-commutative groupoids obtained from the failure of 3-uniqueness in stable theories

We construct a possibly non-commutative groupoid from the failure of $3$-uniqueness of a strong type. The commutative groupoid constructed by John Goodrick and Alexei Kolesnikov in \cite{GK} lives in the center of the groupoid. A certain automorphism approximated by the vertex groups of the non-commutative groupoids is suggested as a fundamental group of the strong type.

Duality for dyadic triangles

This paper is a direct continuation of the paper “Duality for dyadic intervals” by the same authors, and can be considered as its second part. Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent groupoid under the binary operation of arithmetic mean. The first paper dealt with the structure of finitely generated subgroupoids of the dyadic line, which were shown to be isomorphic to dyadic intervals. Then a duality between the class of dyadic intervals and the class of certain subgroupoids of the dyadic unit square was described. The present paper extends the results of the first paper, provides some characterizations of dyadic triangles, and describes a duality for the class of dyadic triangles. As in the case of intervals, the duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the dyadic unit cube, considered as (commutative, idempotent and entropic) groupoids with additional constant operations.

Duality for dyadic intervals

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of (real) polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. This paper is a first step in finding a duality for dyadic polytopes, analogues of real convex polytopes, but defined over the ring [Formula: see text] of dyadic rational numbers instead of the ring of reals. A dyadic [Formula: see text]-dimensional polytope is the intersection with the dyadic space [Formula: see text] of an [Formula: see text]-dimensional real polytope whose vertices lie in the dyadic space. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polytopes carry the structure of a commutative, entropic and idempotent groupoid under the operation of arithmetic mean. Such dyadic polytopes do not preserve all properties of real polytopes. In particular, there are infinitely many (pairwise non-isomorphic) dyadic intervals. We first show that finitely generated subgroupoids of the groupoid [Formula: see text] are all isomorphic to dyadic intervals. Then, we describe a duality for the class of dyadic intervals. The duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the square of the dyadic unit interval with additional constant operations. A second paper deals with a duality for dyadic triangles.

The $x$-divisor pseudographs of a commutative groupoid

The notion of a zero-divisor graph is considered for commutative groupoids with zero. Moufang groupoids and certain medial groupoids with zero are shown to have connected zero-divisor graphs of diameters at most four and three, respectively. As $x$ ranges over the elements of a commutative groupoid $\mB$ (not necessarily with zero), a system of pseudographs is obtained such that the vertices of a pseudograph are the elements of $\mB$ and vertices $a$ and $b$ are adjacent if and only if $ab=x$. These systems are completely characterized as being partitions of complete pseudographs $\overline{K}_{n}$ whose parts are indexed by the vertices of $\overline{K}_{n}$. Furthermore, morphisms are defined in the class of all such systems of pseudographs making it (categorically) isomorphic to the category of commutative groupoids, thereby combinatorializing the theory of commutative groupoids. Also, concepts of ``congruence" and ``direct product" that are compatible with those in the category of commutative groupoids are established for these systems of pseudographs.

Weak subgroupoid lattices

Abstract The aim of the paper is a characterization of the lattice of all weak subgroupoids of a partial groupoid. It also extends to arbitrary finite algebras Pióro’s result saying that the weak subgroupoid lattice of a finite commutative groupoid G in which g·h ≠ g (for all g, h ∈ G) uniquely determines its subgroupoid lattice.

RECONSTRUCTING MULTISETS OVER COMMUTATIVE GROUPOIDS AND AFFINE FUNCTIONS OVER NONASSOCIATIVE SEMIRINGS

A reconstruction problem is formulated for multisets over commutative groupoids. The cards of a multiset are obtained by replacing a pair of its elements by their sum. Necessary and sufficient conditions for the reconstructibility of multisets are determined. These results find an application in a different kind of reconstruction problem for functions of several arguments and identification minors: classes of linear or affine functions over nonassociative semirings are shown to be weakly reconstructible. Moreover, affine functions of sufficiently large arity over finite fields are reconstructible.

Definability for equational theories of commutative groupoids

We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.

Commutative medial ternary groupoids

The paper studies the class of commutative medial ternary groupoids. A construction of ternary semiterms is given and it is proved that the equational theory of medial commutative ternary groupoids is solvable, namely, an algorithm is found, which in allmedial commutative ternary groupoids verifies the validity of the identity u = v for any pair (u, v) of terms. A construction of free medial commutative ternary groupoids is given, and it is proved that anymedial commutative ternary groupoid has a convex linear representation.

Algebraic Structure of TCI-Groupoids: Classes, Clones, Constructions

Some aspects of the algebraic structure of TCI-groupoids (totally commutative idempotent groupoids) are presented. Among other things, a full description of the corresponding binary clones is given. To investigate the algebraic structure of TCI-groupoids and their clones the families of two-generated subgroupoids are used. Next the algebraic structure of these families is studied in detail. We present a special construction which allows us to build each finite TCI-groupoid using the two-element semilattice as a kind of elementary "brick". Furthermore it is proved that the class of all regular TCI-groupoids satisfying binary nonbalanced identities is a nonaxiomatizable proper subclass of the class of all regular TCI-groupoids.

The ordering of commutative terms

By a commutative term we mean an element of the free commutative groupoid F of infinite rank. For two commutative terms a, b write a ⩽ b if b contains a subterm that is a substitution instance of a. With respect to this relation, F is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other things, we prove that every commutative term (or its block in the factor) is a definable element. Consequently, the ordered set has no automorphisms except the identity.

Theorems of Idempotent Commutative Groupoids

In this paper, among other results, we prove that a clone C with five essentially binary operations is minimal if and only if C is a clone of a non-trivial affine space over GF(7). This result is a product of systematic investigation of varieties of idempotent commutative groupoids. It is probably the most spectacular result of this investigation, but many other results formulated and proved throughout the paper may turn out more useful in further study in this area.

Numbers implying algebraic structures

We prove in this paper that a commutative idempotent groupoid is a proper Plonka sum of affine spaces over GF(3) if and only if it has 27 essentially 4-ary term functions. This means, in another terminology, that the number 27 is the characteristic number of those sums in the variety of all commutative idempotent groupoids. Using this result, we also show that a medial idempotent groupoid with three essentially binary operations contains a subgroupoid term equivalent to either the five-element affine space over GF(5) or a Plonka sum of a trivial groupoid and the three-element affine space over GF(3).

Group-Like Structure Underlying the Unit Ball in Real Inner Product Spaces

Abstraction of the relativistic velocity addition law and of the Thomas rotation of the special theory of relativity yields a means of endowing the unit ball in any real inner product space with a group- like structure, in which the standard associative- commutative laws are relaxed by means of the Thomas rotation. The resulting group- like object is called a complete weakly associative- commutative groupoid. Any complete WACG can be extended to a group analogous to the Lorentz group of the special theory of relativity.

Decompositions of universal algebras by idempotent algebras

An algebra with units is an algebra in which every subalgebra contains a singleton subalgebra. A one-unit-algebra is an algebra in which every subalgebra contains exactly one singleton subalgebra. IfU,V are subclasses of a classK of algebras,UKV is the class of all\(\mathfrak{A}\)eK on which there is a congruence θ such that\(\mathfrak{A}\)/θeV and every θ-class that is a subalgebra of\(\mathfrak{A}\) belonging toK belongs also toU, e.g., ifK is the class of all semigroups,V is the class of all bands andU is the class of all groups,UKV is the class of all bands of groups. We studyUKV andUKU whereU is a class of one-unit-K-algebras andV is a class of idempotentK-algebras. IfK is a class of algebras of type τ closed under subalgebras and homomorphisms,U is the class of all one-unit-K-algebras andV is the class of all idempotentK-algebras, thenUKV is the class of allK-algebras that are τ-reducts of 〈τ, e〉-algebras\(\mathfrak{A}\) satisfying e(x) is a singleton subalgebra for everyx e A belonging to the τ-subalgebra of\(\mathfrak{A}\) generated byx and e(f(− x1, x2,..., xn))=e(fe(x1), e(x2),..., e(xn)) for every n-ary operationf of type τ. IfK is a variety of algebras with units and of finite type,U andV are finitely based (relative toK) subquasivarieties ofK, thenUKV is finitely based relative toK. IfK is the variety of all commutative groupoids with an additional unary operatione satisfying e(e(x))=e(x)=e(x)· e(x), e(x · y)=e(x)· e(y),U andV are the subvarities ofK defined by e(x)=e(y) andx=e(x) respectively, thenUKU is neither a variety nor finitely based. Some applications to semigroups and quasigroups are considered.

On the Structure of Equationally Complete Varieties. II

Each member $\mathcal {V}$ of a large family of nonassociative or, when applicable, nondistributive varieties has the following universal property: Every variety $\mathcal {K}$ that satisfies certain very weak versions of the amalgamation and joint embedding properties is isomorphic, as a category, to a coreflective subcategory of some equationally complete subvariety of $\mathcal {V}$. Moreover, the functor which serves to establish the isomorphism preserves injections. As a corollary one obtains the existence of equationally complete subvarieties of $\mathcal {V}$ that fail to have the amalgamation property and fail to be residually small. The family of varieties universal in the above sense includes commutative groupoids, bisemigroups (i.e., algebras with two independent associative operations), and quasi-groups.

On the structure of equationally complete varieties. II

Each member $\mathcal {V}$ of a large family of nonassociative or, when applicable, nondistributive varieties has the following universal property: Every variety $\mathcal {K}$ that satisfies certain very weak versions of the amalgamation and joint embedding properties is isomorphic, as a category, to a coreflective subcategory of some equationally complete subvariety of $\mathcal {V}$. Moreover, the functor which serves to establish the isomorphism preserves injections. As a corollary one obtains the existence of equationally complete subvarieties of $\mathcal {V}$ that fail to have the amalgamation property and fail to be residually small. The family of varieties universal in the above sense includes commutative groupoids, bisemigroups (i.e., algebras with two independent associative operations), and quasi-groups.

Testing Categories and Strong Universality

A category A is binding (or universal) if any full category of algebras is isomorphic to a full subcategory of A. There are many binding categories: the category of all commutative rings with unit and all unit-preserving homomorphisms [1], the category of bounded lattices [2], the category of semigroups [3], the category A(1, 1) of all algebras with two unary fundamental operations and the category of directed graphs [4], the category of all commutative groupoids [11] and many others.