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Elements of Algebra
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
If a set X has an identity element e with respect to a binary operation ∘, then an element y∈X is called an inverse of x∈X provided that x°y=y°x=e. Note that the set of all real-valued n×n matrices under matrix multiplication has a multiplicative identity, namely the n×n identity matrix. Nonetheless, not every n×n matrix has a multiplicative inverse.
Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
The set of all integers forms an Abelian group with respect to the operation of addition. Clearly, the sum of two integers is an integer, addition is associative, and the identity element can be taken as 0 since the addition of 0 to any integer does not alter it. The inverse of any integer is then the negative of the integer since a+(-a)=0 $ a + (- a) = 0 $ . Groups whose basic operation is addition are sometimes called additive groups.
Elements of Galois Fields
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
To see why this is the case, we first assume that p is a prime number. Associativity and commutativity of modulo-p multiplication are obvious. The identity element is 1. ∈ is closed since there are no m, n ∈ G such that m · n = 0 modulo-p (otherwise p must divide m · n, but p and m · n are relatively prime since m, n < p and p is prime). Furthermore, every element of G has an inverse (see Problem 4.6.2). Therefore, G is a finite group under modulo-p multiplication.
A rough set model based on (L, M)-fuzzy generalized neighborhood systems: a constructive approach
Published in International Journal of General Systems, 2022
Kamal El-Saady, Hussein S. Hussein, Ayat A. Temraz
A semi-quantale is said to be unital (Rodabaugh 2007) if the binary operation ⊗ has an identity element called the unit. If the unit e of the groupoid coincides with the top element of L, then a unital semi-quantale is called a strictly two-sided or integral semi-quantale.commutative (Rodabaugh 2007) if the binary operation ⊗ is commutative, i.e. , .a quantale (Rosenthal 1990) if the binary operation ⊗ is associative and satisfies
Fifty years of similarity relations: a survey of foundations and applications
Published in International Journal of General Systems, 2022
Let X be a set, an ordered semigroup with identity element e and m a map . is called a generalized metric space and m a generalized metric on X if and only if for all
A review on some classes of algebraic systems
Published in International Journal of Control, 2020
Víctor Ayala, Heriberto Román-Flores
In this review, we consider a semigroup associated with some classes of algebraic control systems. Means, systems with an algebraic structure on the manifold and the dynamics. It turns out that on a connected Lie group, a semigroup with a nonempty interior containing a neighbourhood of the identity element e generates all the group. Therefore, when the accessible set from the identity is a semigroup, local controllability from e implies global controllability from the identity. Unfortunately, in Ayala and SanMartin (2001), it is shown that for a linear control system on the Lie group the accessibility set from e is not a semigroup. Furthermore, for a transitive linear system on a Lie group G, Jouan (2011) shows that is a semigroup if and only if it coincides with G.