Tolerance in algebraic structures
Bohdan Zelinka · Czechoslovak Mathematical Journal · 1970
ZEEMAN [3] introduces the concept of tolerance on a set as a reflexive and symmetric relation.M. A. ARBIB [1,2] applies this concept in the theory of automata, B. ZELINKA [4] in the theory of graphs.Here we shall introduce this concept into abstract algebra.As mentioned above, the tolerance is a reflexive and symmetric relation on a set.If on a set M a tolerance £, is given, we speak about the tolerance space (M, 0-Now let an algebraic structure 91 = (A, ^) be given.(By the symbol A we denote the set of elements of the algebraic structure, by the symbol ^ the set of operations on this set.)On the set A let a tolerance ^ be given.We say that ЭД is a (^-tolerance algebraic structure, if and only if the following holds: Let / G #" and let/ be an n-ary operation.If we have In elements x^, ..., x", ji, ...,};" of Л such that (x^-, y) e ^ for i == 1, ..., n, then also (/(xi, ..., x"),/(j;i, ..., y"))e(^.We shall investigate the most important types of algebraic structures -groups, semigroups, rings, fields and lattices.L GROUPS Theorem 1.Let G be a group, let a tolerance ^ be given on its set of elements.If G is a ^-tolerance semigroup with respect to its multiplication, it is also a ^-tolerance group.Proof.The fact that G is a (^-tolerance semigroup means that (x^, yj e ^, (x2, У2) ^ e ^ imphes (xiX2, У1У2) ^ ^ for arbitrary elements Xj, X2, Уи у г of G. To prove that G is a (^-tolerance group it is necessary and sufficient to prove that (x, y) e с implies (x"^, y~^) e ^ for arbitrary elements x, у of G.