Associative Binary Operations on a Set with Four Elements

The 16th Annual International Symposium on Computer Architecture, 1989

The traditional approach to implementing wide setassociativity is expensive, requiring a wide tag memory (directory) and many comparators. Here we examine altemative implementations of associativity that use hardware similar to that used to implement a direct-mapped cache. One approach scans tags serially from most-recently used to least-recently used. Another uses a partial compare of a few bits from each tag to reduce the number of tags that must be examined serially. The drawback of both approaches is that they increase cache access time by a factor of two or more over the traditional implementation of setassociativity, making them inappropriate for cache designs in which a fast access time is crucial (e.g. level one caches, caches directly servicing processor requests).

International Journal of Intelligent Computing Research

The paper is focused on some theoretical questions of the Database Theory. Multiset table algebra is considered. The notion of a table specified using the notion of a multiset (or bag). A signature of multiset table algebra is filled up with new operations such as inner and outer joins, semi-join and aggregate operations. A formal mathematical semantics of these operations is defined. The special element NULL is inserted in the universal domain for define of the outer join.

Applied Sciences

Some results regarding the semi-cyclicity, cyclicity, semi-commutativity and commutativity of certain n-ary operations defined on Cartesian powers provided by special algebraic structures are given. The obtained results are extensions of some previous results of the first author, A. M. Gal’mak. The paper provides many examples of algebraic n-ary structures.

European Journal of Combinatorics, 1985

A subset R,; 53 will be called (associatively) admissible if there exists a binary operation * defined on 5 such that x*(y*z)=(x*y)*z iff (X,Y,Z)ER.

2015

In this work, we study the elliptic curve over the ring ; ; where d is a positive integer. More precisely in cryptography applications, we will give many various explicit formulas describing the binary operations calculus in . The motivation for this work came from the observation that several practical discrete logarithm-based cryptosystems, such as ElGamal, the Elliptic Curve Cryptosystems. Keywords— Elliptic Curves, Finite Ring, Cryptography..

arXiv: Group Theory, 2017

In this paper we introduce novel views of monoids and groups. More specifically, for a given set $S$, let $S^{S\times S}$ be the set of binary operations on $S$. We equip $S^{S\times S}$ with canonical binary operations induced by the elements of $S$. Let $S^{S\times S}_{mn}$ (respectively, $S^{S\times S}_{gr}$) be the set of binary operations that make $S$ monoids (respectively, groups). Then we have the following "duality": for each $z\in S^{S\times S}_{mn}$ a certain subset of $S^{S\times S}$, denoted by $S^*_z$, is a monoid with a canonical binary operation and is isomorphic to $(S,z)$. If $z\in S^{S\times S}_{gr}$, then $S^{S\times S}_{gr}$ can be partitioned into copies of $S^*_z$. We also give a new characterization of group binary operations which distinguishes them from the other binary operations. These results give us new insights into monoids and groups, and will provide new tools and directions in studying these objects.

2020

A binary operation is a calculus that combines two elements to obtain another elements. It seems quite simple for numbers, because we usually imagine it as a simple sum or product. However, also in the case of numbers, a binary operation can be extremely fascinating if we consider it in a generalized form. Here the reader can find several examples of generalized sums for different sets of numbers (Fibonacci, Mersenne, Fermat, q-numbers, repunits and many other numbers). These sets can form groupoid which possess different binary operators. As we will see at the end of this exposition of cases, the most relevant finding is that different integer sequences can have the same binary operator and that, consequently, can be used as different representations of the same groupoid.

2015

Given a multiplicative band of idempotents S in a ring R, for all e, f ∈ S the ∇-product e∇f = e + f + f e − ef e − f ef is an idempotent that lies roughly above e and f in R just as ef and f e lie roughly below e and f. In this paper we study ∇-bands in rings, that is, bands in rings that are closed under ∇, giving various criteria for ∇ to be associative, thus making the band a skew lattice. We also consider when a given band S in R generates a ∇-band.

2010

Properties like commutativity and associativity are very important in many applications. To make detecting these properties easier, it is desirable to reformulate them in the simplest possible way. A known way to simplify these descriptions is to use postfix notations, in which there is no need to use parentheses. We show that some of these properties can be further simplified if we represent them as equalities not between well-defined expressions, but as equalities between expression fragments.