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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.
Algebraic Structures
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
We now consider some definitions which will enable us to distinguish the properties of certain binary operations. The first relates to the fact that a binary operation * combines pairs of elements of a set, so that there are two different readings which we could give to the expression a*b*c. We could interpret it as (a*b)*c, i.e. combine a with b first and then combine the result with c. Alternatively, we could perform the operations in the order a*(b*c), combining a with the result of b*c. For some binary operations, for example subtraction on the real numbers, the two interpretations give different results. For others, such as addition, the method of grouping the three elements makes no difference. Binary operations which have the latter property are termed ‘associative'.
A New Metatheorem and Subdirect Product Theorem for L-Subgroups
Published in Fuzzy Information and Engineering, 2018
It can be established that is an injective function. Also, is an order reversing function from J into . The ordering of induces a pointwise partial order on . That is for P and Q in Further as is a lattice, is also a lattice. Tom Head enlarged the domain of a binary operation on G to by convolutional extension method. Also, the binary operation on G induces a binary operation on and .
An alternative construction of uninorms on bounded lattices
Published in International Journal of General Systems, 2023
Let , be an interior operator and be a t-conorm. Then the binary operation given by the formula (7) is a uninorm on with the identity element e.
A Profit Maximisation Solid Transportation Problem Using Genetic Algorithm in Fuzzy Environment
Published in Fuzzy Information and Engineering, 2021
S. Samanta, A. Ojha, B. Das, S. K. Mondal
Let be a binary operation on the set of real numbers. If and are two closed intervals then defines a binary operation on the set of closed intervals. In the case of division, it is assumed that 0 B.