In set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, a semiset is a proper class

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...

 which is contained in a set.
The theory of semisets was proposed and developed by Czech

Czech Republic

The Czech Republic is a landlocked country in Central Europe. The country is bordered by Poland to the northeast, Slovakia to the east, Austria to the south, and Germany to the west and northwest....

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

s Petr Vopěnka

Petr Vopenka

Petr Vopěnka is a Czech mathematician. In the early seventies, he established the Alternative Set Theory , which he subsequently developed in a series of articles and monographs...

 and Petr Hájek

Petr Hájek

Petr Hájek is a Czech scientist in the area of mathematical logic and a professor of mathematics. He works at the Institute of Computer Science at the Academy of Sciences of the Czech Republic and worked as a lecturer at the Faculty of Mathematics and Physics at the Charles University in Prague...

 (1972). It is based on a modification of the von Neumann-Bernays-Gödel set theory; in standard NBG, the existence of semisets is precluded by the axiom of separation. The concept of semisets opens the way for a formulation of an alternative set theory

Alternative set theory

Generically, an alternative set theory is an alternative mathematical approach to the concept of set. It is a proposed alternative to the standard set theory.Some of the alternative set theories are:*the theory of semisets...

.

Semisets can be used to represent sets with imprecise boundaries. Vilém Novák (1984) studied approximation of semisets by fuzzy set

Fuzzy set

Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced simultaneously by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to...

s, which are often more suitable for practical applications of the modeling of imprecision.