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Mathematical Foundations
Published in Chintan Patel, Nishant Doshi, Internet of Things Security, 2018
Remember:If (S, △, ◻, I) is group and ′ (S′, △′, ◻′, I′) is sub-algebra of then ′ is also called as a subgroup of . So “a sub-algebra of group is a group”.Order of the group can be represented by ||. || is total number of elements in the set defined over group. Order of finite group is the finite integer number.A group is a monoid in which every element is invertible.
Modelling Manufacturing Systems in a Dioid Framework
Published in Javier Campos, Carla Seatzu, Xiaolan Xie, in Manufacturing, 2018
Thomas Brunsch, Laurent Hardouin, Jörg Raisch
Definition 2.8 (Monoid) A monoid, (ℳ,⋅,e), is a set M endowed with an internal binary operation ⋅, which is associative, and with an identity element e. If the internal law ⋅ is commutative, (ℳ,⋅,e) is said to be a commutative monoid. If the internal law ⋅ is idempotent, that is, a⋅a=a∀a∈ℳ, the monoid is said to be idempotent.
Parameterized simplification logic I: reasoning with implications and classes of closure operators
Published in International Journal of General Systems, 2020
Pablo Cordero, Manuel Enciso, Angel Mora, Vilem Vychodil
Throughout this paper, we are going to utilize lattice-based structures that are closely related to structures frequently used in multiple-valued logics (Galatos et al. 2007; Gottwald 2008; Hájek 1998). Namely, we are going to use complete dual residuated lattices. A complete dual residuated lattice is a structure satisfying the following conditions: is a complete lattice where 0 is the least element and 1 is the greatest element. As usual, we use the symbols ∨ and ∧ to denote suprema (least upper bounds) and infima (greatest lower bounds), respectively; is a commutative monoid;⊖ is a binary operation so that the pair satisfies the following adjointness property: For all , we have
Ideals of an EMV-semiring
Published in International Journal of General Systems, 2020
R. A. Borzooei, M. Shenavaei, A. Di Nola, O. Zahiri
An algebra of type is called a semiring, if it satisfies the following conditions: (S1) is a commutative monoid with identity 0,(S2) is a semigroup,(S3) multiplication distributes over addition,(S4) for all , .
Modal translation of substructural logics
Published in Journal of Applied Non-Classical Logics, 2020
An FL-algebra is a structure where is a bounded lattice is a partially ordered monoid (○ is monotone and associative and is a two-sided identity element ) are residuated, i.e. iff iff for any ,