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Vagueness and its representations: a unifying look.

Maciej Wygralak (1998)

Mathware and Soft Computing

Using the notion of a vaguely defined object, we systematize and unify different existing approaches to vagueness and its mathematical representations, including fuzzy sets and derived concepts. Moreover, a new, approximative approach to vaguely defined objects will be introduced and investigated.

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...

Vector sets with no repeated differences

Péter Komjáth (1993)

Colloquium Mathematicae

We consider the question when a set in a vector space over the rationals, with no differences occurring more than twice, is the union of countably many sets, none containing a difference twice. The answer is “yes” if the set is of size at most 2 , “not” if the set is allowed to be of size ( 2 2 0 ) + . It is consistent that the continuum is large, but the statement still holds for every set smaller than continuum.

Very small sets

Haim Judah, Amiran Lior, Ireneusz Recław (1997)

Colloquium Mathematicae

Vitali sets and Hamel bases that are Marczewski measurable

Arnold Miller, Strashimir Popvassilev (2000)

Fundamenta Mathematicae

We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null. The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first. We show that there is no easy way to get a Marczewski null Hamel basis for the reals from one...

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