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Valuations of lines

Josef Mlček — 1992

Commentationes Mathematicae Universitatis Carolinae

We enlarge the problem of valuations of triads on so called lines. A line in an e -structure 𝔸 = A , F , E (it means that A , F is a semigroup and E is an automorphism or an antiautomorphism on A , F such that E E = 𝐈𝐝 A ) is, generally, a sequence 𝔸 B , 𝔸 U c , c 𝐅𝐙 (where 𝐅𝐙 is the class of finite integers) of substructures of 𝔸 such that B U c U d holds for each c d . We denote this line as 𝔸 ( U c , B ) c 𝐅𝐙 and we say that a mapping H is a valuation of the line 𝔸 ( U c , B ) c 𝐅𝐙 in a line 𝔸 ^ ( U ^ c , B ^ ) c 𝐅𝐙 if it is, for each c 𝐅𝐙 , a valuation of the triad 𝔸 ( U c , B ) in 𝔸 ^ ( U ^ c , B ^ ) . Some theorems on an existence of...

-representation and set-prolongations

Josef Mlček — 1992

Commentationes Mathematicae Universitatis Carolinae

By an -representation of a relation we mean its isomorphic embedding to 𝔼 = { x , y ; x y } . Some theorems on such a representation are presented. Especially, we prove a version of the well-known theorem on isomorphic representation of extensional and well-founded relations in 𝔼 , which holds in Zermelo-Fraenkel set theory. This our version is in Zermelo-Fraenkel set theory false. A general theorem on a set-prolongation is proved; it enables us to solve the task of the representation in question.

Monotonic valuations of π σ -triads and evaluations of ideals

Josef Mlček — 1993

Commentationes Mathematicae Universitatis Carolinae

We develop problems of monotonic valuations of triads. A theorem on monotonic valuations of triads of the type π σ is presented. We study, using the notion of the monotonic valuation, representations of ideals by monotone and subadditive mappings. We prove, for example, that there exists, for each ideal J of the type π on a set A , a monotone and subadditive set-mapping h on P ( A ) with values in non-negative rational numbers such that J = h - 1 ' ' { r Q ; r 0 & r 0 } . Some analogical results are proved for ideals of the types σ , σ π and...

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