Remarks on Q-independence of Stone algebras
by
Anna Chwastyk
Technical University of Opole
Coauthors: Kazimierz Glazek (University of Zielona Gora)

In 1958 E. Marczewski introduced a general notion of independence, which contained as special cases the majority of independence notions used in various branches of mathematics. A non-empty set X of the carrier A of an algebra A = (A; F) is called M-independent if equality of two term operations f and g of the algebra A on any finite system of different elements of X implies f=g in A. There are several interesting results on this notion of independence. However the important scheme of M-independence is not wide enough to cover stochastic independence, independence in projective spaces and some others. This is why some notions weaker than the M-independence have been developed. The notion of independence with respect to a family Q of mappings (defined on subsets of A) into A, Q-independence for short, is a common way of defining almost all known notions of independences. A non-empty set X of the carrier A of an algebra A is called Q-independent if equality of two term operations f and g of the algebra A on any finite system of elements a₁, a₂, ..., a_n of X implies f(p(a₁), p(a₂), ..., p(a_n)) = g(p(a₁), p(a₂), ..., p(a_n)) for any mapping p in Q.

Recall that an algebra L=(L; \/ , /\ , ^*, 0, 1) of type (2, 2, 1, 0, 0) is said to be a Stone algebra if (L; \/ , /\ , 0, 1) is a bounded distributive lattice, ^* is unary operation on L such that a /\ x=0 iff x <= a^*, and the so-called Stone identity x^* \/ x^**=1 holds. We investigate Q-independence subsets in Stone algebras for some specified families Q of mappings (e.g. M, S, S₀, and A₁, see [2], p. 122), using the well-known triple representation of Stone algebras.

References. [1] K. Glazek, Independence with respect to a family of mappings, Dissertationes Math. 81 (1971), 1-55.
[2] K. Glazek, General notions of independence, p. 112-128 in Advances in Algebra, World Scientific, Singapore 2003.
[3] E. Marczewski, Independence with respect to a family of mappings, Colloq. Math. 20 (1968), 11-17.

Date received: May 19, 2004