Abstract: A method for efficiently comparing two trinary logic representations, including the steps of creating a first data structure (a VALUE data structure) representative of a first set of properties; creating a second data structure (a KNOWN data structure) representative of whether the first set of properties is known; creating a third data structure (a TARGET data structure) representative of a target set of properties; creating a fourth data structure (a WANT data structure) representative of whether the target set of properties is wanted; and comparing the first, second, third, and fourth data structures using bit-wise binary operations to determine whether the first set of known properties are wanted as a target set of properties. In exemplary embodiments, the bit-wise binary operations are performed according to the Boolean equation: (not WANT) or (KNOWN and ((TARGET xor VALUE))). Alternatively, the bit-wise binary operation are performed according to the Boolean equation: (not WANT) or (KNOWN and ((TARGET and VALUE) or ((not TARGET) and (not (VALUE))). These data structures may be any size computer word, including 16 and 32-bit words.

Abstract: We study the various term operations on the set of skew primitive elements of Hopf algebras, generated by skew primitive semi-invariants of an Abelian group of grouplike elements. All 1-linear binary operations are described and trilinear and quadrilinear operations are given a detailed treatment. Necessary and sufficient conditions for the existence of multilinear operations are specified in terms of the property of particular noncommutative polynomials being linearly dependent and of one arithmetic condition. We dub the conjecture that this condition implies, in fact, the linear dependence of the polynomials in question and so is itself sufficient (a proof of this conjecture see in "An existence condition for multilinear quantum operations," Journal of Algebra, 217, 1999, 188-228).

Abstract: A (finite) acyclic connected graph is called a tree. Let W be a finite nonempty set, and let M(W) be the set of all trees T with the property that W is the vertex set of T. We will find a one-to-one correspondence between H(J^) and the set of all binary operations on W which satisfy a certain set of three axioms (stated in this note).

Abstract: This paper shows that the collection of identities in two variables which hold in the algebra N of the natural numbers with constant zero, and binary operations of sum and maximum does not have a finite equational axiomatization. This gives an alternative proof of the non-existence of a finite basis for N-a result previously obtained by the authors.

Abstract: A commutative pomonoid is a structure A = 〈A;⊕, 0;≤〉, whose reduct 〈A;⊕, 0〉 is a commutative monoid where ≤ is a partial order of A for which ⊕ is isotone in both of its arguments. We call A residuated provided that for any x, y ∈ A there is a least z ∈ A such that x ≤ z ⊕ y, in which case this z is denoted by x . − y and the binary operation . − on A is called residuation. In particular, such structures A satisfy x ≤ y ⇔ x . − y ≤ 0. The abstract study of such pomonoids was inspired by the ideal lattices of commutative unital rings, with ideal multiplication, reversed set inclusion and the ring itself in the roles of ⊕, ≤ and 0. Here, although (unary) inverses are absent, residuation supplies a binary operation abstracting division. More recently, residuated commutative pomonoids in their full generality have received attention as natural models for fragments of lin-

Abstract: A system is provided for finding the result of a binary operation performed on an array of values, along with its address, in a storage-efficient manner. The system is based on a binary tree structure having a pipeline of binary operators and corresponding multiplexers and storage elements to store the outputs of the multiplexers in addition to the partial addresses of the outputs represented by the outputs of the binary operators. Each computation stage adds one more bit of address until the address is completely known at the last computation stage. The invention reduces the amount of storage required for keeping track of the addresses that are the result of the binary operation.

Abstract: For universal hyperalgebras we study a binary operation of exponentiation which to a pair of universal hyperalgebras of a given type assigns their power, i.e., a universal hyperalgebra of the same type carried by the corresponding set of homomorphisms. We give sufficient conditions for the existence of such a power and for a decent behaviour of the exponentiation. As a consequence of our investigations we discover a cartesian closed subcategory of the category of universal hyperalgebras of the same type and homomorphisms between them.

Abstract: Cut-and-project sets with convex acceptance windows, based on irrationalities τ=\( \frac{1}{2} \)(1+√5), β=1+√2, μ=2+√3 are models for experimentally observed quasicrystals – materials with diffraction patterns consisting of sharp Bragg peaks in crystallographically disallowed patterns. We show that for each of these three irrationalities there exists a unique binary operation of the type x⊢sy:=sx+(1−s)y, such that one-dimensional cut-and-project sets are precisely Delone sets closed under this operation.

Abstract: A notion of a right (left) crossed homomorphism of finite algebras with a scheme of binary operators is introduced. This notion generalizes the notion of a right (left) crossed isotopy of quasigroups introduced by V.D. Belousov. A theorem on crossed homomorphisms (an analogue of the classical theorem on homomorphisms) is proved. The description of crossed homomorphisms of an algebra with a scheme of operators is reduced to the description of its crossed congruences. Crossed congruences of quasigroups that are isotopic to groups and cross-isotopic to groups are studied. The possibility of applying crossed congruences to constructing algorithms for solving equations over algebras is shown.

Abstract: This note is an incomplete summary of results on single equational axioms for algebraic theories Pioneering results were obtained decades ago without the use of computers by logicians such as Tarski Higman Neumann and Padmanabhan Use of today s high speed computers and sophisticated soft ware for searching for proofs and counterexamples has led to many additional results Considered here are equationally de ned algebras and the goal is to nd simple single equations bases that axiomatize algebras that are ordinarily presented with sets of equations For example a standard way to de ne group theory is as an algebra with a binary operation a unary operation and a constant e satisfying the three equations

Abstract: In this we study a wide class of optimal path problem in acyclic digraphs, where path lengths are defined through associative binary operations:addition, multiplication, multiplication-addition, fraction and so on. Solving a system of two interrelated recur-sive equations, we simultaneously find both shortest and longest path lengths, Further, for every problem (primal problem), we associate the other related problem (negative-equivalent problem) where each path length is defined through the associative operation connected to it in the primal problem by DeMorgan’s law. The main objective of this paper is to derive a negative-equivalency theorem between the primal problem and the negative-equivalent one

Abstract: This chapter summarizes main ways to extend classical set-theoretic operations (complementation, intersection, union, set-difference) and related concepts (inclusion, quantifiers) for fuzzy sets Since these extensions are mainly pointwisely defined, we review basic results on the underlying unary or binary operations on the unit interval such as negations, t-norms, t-conorms, implications, coimplications and equivalences Some strongly related connectives (means, OWA, weighted, and prioritized operations) are also considered, emphasizing the essential differences between these and the formerly investigated operator classes We also show other operations which have no counterpart in the classical theory but play some important role in fuzzy sets (like symmetric sums, weak t-norms and conorms, compensatory AND)

Abstract: We consider the process theory PA that includes an operation for parallel composition, based on the interleaving paradigm. We prove that the standard set of axioms of PA is not ω-complete by providing a set of axioms that are valid in PA, but not derivable from the standard ones. We prove that extending PA with this set yields an ω-complete specification, which is finite in a setting with finitely many actions.

Abstract: This paper shows that the collection of identities in two variables which hold in the algebra N of the natural numbers with constant zero, and binary operations of sum and maximum does not have a finite equational axiomatization. This gives an alternative proof of the nonexistence of a finite basis for N--a result previously obtained by the authors.

Abstract: In this paper we describe conditional probabilities as difference set. The main idea is that a system is in same state and from this state is can get to another state if there are fulfield some properties. In other words we have a partial binary operation ⊖ such that the operation ⊖ can be interpreted as a change of a state. If we put some logical questions about change of states on this we get the structure which is called a difference set.