Non-associative Operators Research Articles

Order from non-associative operations

Algebraic structures are often converted to ordered structures to gain information about the algebra using the properties of partially ordered sets. Such studies have been predominantly undertaken for semigroups, using various proposed relations. This has led to a spate of works dealing with associative fuzzy logic connectives (FLCs) and the orders that they generate. One such relation, proposed by Clifford, is employed both for its generality as well as utility. In a recent work, Gupta and Jayaram classified the semigroups that yield a partial order through the relation. In this work, we characterise groupoids that would give a partial order by introducing a property called the Generalised Quasi-Projectivity. Further, for the groupoids that lead to an ordered set, we explore the monotonicity of the underlying groupoid operation on the obtained poset. Finally, in light of the above results, we explore the major non-associative fuzzy logic connectives along these lines, thus complementing and augmenting, already existing works in the literature. Our work also shows when an FLC from a given class of operations remains one even w.r.to the order generated from it.

On Fuzzy Implications Derived from General Overlap Functions and Their Relation to Other Classes

There are distinct techniques to generate fuzzy implication functions. Despite most of them using the combination of associative aggregators and fuzzy negations, other connectives such as (general) overlap/grouping functions may be a better strategy. Since these possibly non-associative operators have been successfully used in many applications, such as decision making, classification and image processing, the idea of this work is to continue previous studies related to fuzzy implication functions derived from general overlap functions. In order to obtain a more general and flexible context, we extend the class of implications derived by fuzzy negations and t-norms, replacing the latter by general overlap functions, obtaining the so-called (GO,N)-implication functions. We also investigate their properties, the aggregation of (GO,N)-implication functions, their characterization and the intersections with other classes of fuzzy implication functions.

Timescale standard to discriminate between hyperbolic and exponential discounting and construction of a nonadditive discounting model

Under the presupposition that human time perception is distorted in intertemporal choice, this study constructs a time scale in the framework of axiomatic measurement. First, the conditions (homogeneity of degree one or two) to identify the form of a time scale are proposed so that one can determine whether the hyperbolic or exponential is a more suitable function for modeling people’s discounting. Homogeneity of degree one implies that subjective time delay is measured by a power scale and its discount function becomes a power-exponential type, whereas homogeneity of degree two implies that subjective time delay is measured by a log scale and its discount function becomes a hyperbolic type. Second, a method is shown for constructing a model that can reflect subadditive and superadditive discounting. A non-commutative and non-associative concatenation operation is provided to generate a time string consisting of subdivided durations that incorporates the effect of repeated delay with a subdivided duration. The discounting model is yielded by substituting these strings in an exponential model equipped with a generalized time scale, and the determination of subadditivity or superadditivity depends on whether the discount function value of a product by the non-commutative and non-associative operation is, respectively, smaller or greater than that of a product by an operation of the usual extensive structure.

Embedding of locally compact Hausdorff topological gyrogroups in topological groups

A topological gyrogroup is a topological space endowed with a compatible nonassociative operation, which shares several common properties with topological groups. In this article, we prove that every locally compact Hausdorff topological gyrogroup G can be embedded in a completely regular topological group Γ as a twisted subset. We also study properties shared by them by proving that G has property ▪ if and only if Γ has property ▪, where ▪ is one of the following properties: connectedness, path connectedness, and separability.

Generalized pseudo-probability measure

A new approach to non-additive set functions is presented. The notion of non-additive measure is extended using non-commutative and non-associative operations with two parameters and the cumulative distribution function of a random variable. One new non-additive measure, generalized pseudo-probability measure, is defined and some of its basic properties are proven.

Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena

To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gomez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions.

One-parameter deformations of the diassociative and dendriform operads

Livernet and Loday constructed a polarization of the nonsymmetric associative operad with one operation into a symmetric operad with two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of , which includes Poisson algebras. We combine this with the dendriform splitting of an associative operation into the sum of two nonassociative operations, and use Koszul duality for quadratic operads, to construct one-parameter deformations of the nonsymmetric dendriform and diassociative operads into the category of symmetric operads.

A type assignment for λ-calculus complete both for FPTIME and strong normalization

One of the aims of Implicit Computational Complexity is the design of programming languages with bounded computational complexity. One of the most promising approaches to this aim is based on the use of lambda-calculus as paradigmatic programming language and the design of type assignment systems for lambda-terms, where types guarantee both the functional correctness and the complexity bound. Along this line, we propose a system of stratified types, inspired by intersection types, where intersection is a non-associative operator. This system is correct and complete for polynomial time when a uniform coding scheme is considered; moreover, all the strongly normalizing terms are typed in it, thus increasing the typing power with respect to the previous proposals, based on Light Logics.

Weaker forms of associativity in orthomodular lattices

Computations in orthomodular lattices are rather difficult because there is no known analogue of the standard disjunctive or conjunctive forms from boolean algebras. Tools admitting simplifications of formulas are rather limited. In previous work, we investigated the possibility of finding associative operations in orthomodular lattices that would admit new techniques of theorem proving. It was found that there are only 6 operations in orthomodular lattices which are (already known to be) associative. In this paper, we replaced associativity by more general properties, in particular by the identities that hold in alternative algebras. We have found that there are 8 nonassociative operations in orthomodular lattices satisfying the identities of alternative algebras. For these operations, we clarified the validity of other similar identities, including the Moufang identities. We hope that this effort can contribute to clarifying to what extent the validity of identities in orthomodular lattices can be algorithmized.

Linear representation of algebras with non-associative operations which are satisfy in the balanced functional equations

Algebras with a pair of non-associative binary operations (f, g) which are satisfy in the balanced quadratic functional equations with four object variables considered. First, we obtain a linear representation for the operations, of this kind of binary algebras (A,f,g), over an abelian group (A, +) and then we generalize the linear representation of operations, to an algebra (A,F) with non-associative binary operations which are satisfy in the balanced quadratic functional equations with four object variables.

The Operadic Modeling of Gauge Systems of the Yang-Mills Type

The basis of operadic variational formalism is presented which is necessary when modeling the operadic systems. A general gauge theoretic approach to the abstract operads, based on the physical measurements concepts, is justified and considered. It is explained how the matrix and Poisson algebra relations can be extended to operadic realm. The tangent cohomology spaces of the binary associative flows with their Gerstenhaber algebra structure can be seen as equally natural objects for operadic modeling, just as the matrix and Poisson algebras in conventional modeling. In particular, the relation of the tangent Gerstenhaber algebras to operadic Stokes law for operadic observables is revealed and discussed. Based on this, the rational (cohomological) variational principle and operadic Heisenberg equation for quantum operadic flows are stated. As a modeling selection rule, the operadic gauge equations of the Yang–Mills type are considered and justified from the point of view of the physical measurements and the algebraic deformation theory. It is also shown how the binary weakly non-associative operations are related to approximate operadic (anti-)self-dual models.

On Tractability and Ideal Problem in Non-Associative Operator Algebras

The question of the existence of non-trivial ideals of Lie algebras of compact operators is considered from different points of view. One of the approaches is based on the concept of a tractable Lie algebra, which can be of interest independently of the main theme of the paper. Among other results it is shown that an infinite-dimensional closed Lie or Jordan algebra of compact operators cannot be simple. Several partial answers to Wojtynski’s problem on the topological simplicity of Lie algebras of compact quasinilpotent operators are also given.

Observables and unobservables in a non-associative quantum theory

It is shown that the non-associative operators in a non-associative quantum theory are unobservables. The observable quantity may be presented only by the elements of some asso- ciative subalgebra. It is shown that the elements of the associative subalgebra are extended objects that can be similar to strings. It is assumed that the non-associative quantum eld theory can be applied to the quantization of strongly interacting elds. The method for obtaining eld equations in a non-associative case is given.

Observables and unobservables in a non-associative quantum theory

It is shown that the non-associative operators in a non-associative quantum theory are unobservables. The observable quantity may be presented only by the elements of some asso- ciative subalgebra. It is shown that the elements of the associative subalgebra are extended objects that can be similar to strings. It is assumed that the non-associative quantum eld theory can be applied to the quantization of strongly interacting elds. The method for obtaining eld equations in a non-associative case is given.

Non-associative slave-boson decomposition

An operator constraint for a slave-boson decomposition in t-J model of high temperature superconductivity is considered. It is shown that the constraint can be resolved by introducing a non-associative operator. In this case the constraint is an antiassociative generating relation of a new algebra. Similar constraint is offered for splitting the gluon.

Non-associative slave-boson decomposition

An operator constraint for a slave-boson decomposition in t-J model of high temperature superconductivity is considered. It is shown that the constraint can be resolved by introducing a non-associative operator. In this case the constraint is an antiassociative generating relation of a new algebra. Similar constraint is oered for splitting the gluon. 2000 MSC: 81Rxx, 17D05

Colorless operators in a non-associative quantum theory

The associators/antiassociators for the product of four non-associative operators are deduced. By analogy with SU ( 3 ) gauge theory the notion of colorless (white) operators is introduced. Some properties of white operators are considered. It is hypothesized that white operators do not give any contribution to corresponding associators/antiassociators. It is suggested that the observables in a non-associative quantum theory correspond to the white operators only.

Towards more realistic (e.g., non-associative) ?and?- and ?or?-operations in fuzzy logic

How is fuzzy logic usually formalized? There are many seemingly reasonable requirements that a logic should satisfy: e.g., since A and B and B and A are the same, the corresponding AND-operation should be commutative. Similarly, since A and A means the same as A, we should expect that the AND-operation should also satisfy this property, etc. It turns out to be impossible to satisfy all these seemingly natural requirements, so usually, some requirements are picked as absolutely true (like commutativity or associativity), and others are ignored if they contradict the chosen ones. This idea leads to a neat mathematical theory, but the analysis of real-life expert reasoning shows that all the requirements are only approximately satisfied. We should require all of these requirements to be satisfied to some extent. We show the preliminary results of analyzing such operations. In particular, we show that non-associative operations explain the empirical 7/spl plusmn/2 law in psychology according to which a person can normally distinguish between no more than 7 plus or minus 2 classes.

On the minimal extension of the sequence $ \langle 0,1,1,7 \rangle $

This note shows that the pn-sequence \( \langle 0,1,1,7 \rangle \) has the minimal extension property in the class of algebras with a nonassociative binary operation. This generalizes the result of J. Galuszka and gives a partial solution of problem 18 raised by G. Gratzer and A. Kisielewicz of whether \( \langle 0,1,1,7 \rangle \) has the minimal extension property. This result reduces the problem to the class of algebras with a semilattice operation.

Characterizations of Boolean algebras of idempotents

The notions of the left (right) Jordan groupoids are introduced. IfR is an associative* ring with the identity and ifU(R) [resp.P(R)] denotes the set of all idempotents (resp. projections) of the* ringR, then the operationsp\(\begin{array}{*{20}c} O \\ 1 \\ \end{array} \)q =p − 2pq − qp + 4qpq andp\(\begin{array}{*{20}c} O \\ 2 \\ \end{array} \)q =q − 2pq − 2qp + 4pqp ifp, q ∈U(R) [resp.p, q ∈P(R)] are the nonassociative linear operations inU(R) [resp. inP(R)]. The present paper shows that the operations\(\begin{array}{*{20}c} O \\ 1 \\ \end{array} \) and\(\begin{array}{*{20}c} O \\ 2 \\ \end{array} \) are associative iffpq=qp forp, q ∈U(R) [resp.p, q ∈P(R)]. As a corollary it follows from this that the orthomodular poset (U(R), ≤, 0, 1,′) is a Boolean algebra [which is commutative, i.e.,pq= qp, p, q ∈U(R)] iff (U(R),\(\begin{array}{*{20}c} O \\ 1 \\ \end{array} \), 0, 1,′) or (U(R),\(\begin{array}{*{20}c} O \\ 2 \\ \end{array} \), 0, 1,′) are Jordan associative groupoids. Similar results hold for (P(R), ≤,0, 1, ′).