By Piergiulio Corsini, Violeta Leoreanu (auth.)
ISBN-10: 1441952454
ISBN-13: 9781441952455
ISBN-10: 1475737149
ISBN-13: 9781475737141
This booklet provides many of the quite a few functions of hyperstructures, specially those who have been came across and studied within the final fifteen years. There are functions to the next matters: 1) geometry; 2) hypergraphs; three) binary family; four) lattices; five) fuzzy units and tough units; 6) automata; 7) cryptography; eight) median algebras, relation algebras; nine) combinatorics; 10) codes; eleven) synthetic intelligence; 12) possibilities.
Audience: Graduate scholars and researchers.
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VI E z. We say that the equivalence relation p is special if {e} is an equivalence class and every product x· fj of equivalence classes is a union of (x, fj)-projective equivalence classes. A quasihypergroup (B,·, e) (where e E B) is called a multivalued loop if e is a scalar identity of Band 'r;/ a E B, there exist unique x, yin B such that e E ax n ya. 51 APPLICATIONS OF HYPERSTRUCTURE THEORY It is easy to verify that a quotient of a loop (A", e) modulo a special equivalence relation p is a multivalued loop (AI p, *{ e} ), where Z E x * fl if and only if Z ~ x· fl.
104. Theorem. V (x, y) E H2, V (u, v) E A(x)xA(y), if U/3KHv then x/3HY. 105. Theorem. If H is a hypergroup, then WKH = K(WH). 106. Theorem. If H is a hypergroup, then: 1) < K H , 0 > is regular if and only if (H, 0) is regular; 2) < K H , 0 > is reversible if and only if (H, 0) is reversible; 3) < KH,D > is feebly quasi-canonical if and only if (H, 0) is feebly quasi-canonical. Hyperrings, hypermodules and vector hyperspaces 107. Definition. < A,+, -,0> where: A (Krasner) hyperring is a hyperstructure 1) (A, +) is a canonical hypergroup; 2) (A, -) is a semigroup endowed with a two-sided absorbing element 0; 3) the product distributes from both sides over the sum.
If o(x) = 00, we say that the characteristic of x is zero and we set: X(x) = O. If o(x) =1= 00, we set X(x) = h, where h is the principal order of x in the canonical hypergroup < A, + > . 110. Definition. ) of X(x) for x E A if it exists and is =1= 0, otherwise we set X(A) = O. Ill. Remark. If y = ax, then X(y) divides X(x). 112. Definition. If < A, +,. > is a hyperring and B is a nonempty subset of A, we say that < B, +,. ). We say that B is a left hyperideal of A if (B, +) is a canonical subhypergroup of A and A .
Applications of Hyperstructure Theory by Piergiulio Corsini, Violeta Leoreanu (auth.)
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