# Contents

## Definition

Given a set $V$ with an extensional relation $\prec$, a proto-union structure is a function $U:V \to V$ such that

• for all elements $a \in V$, $b \in V$, and $c \in V$, if $a \prec b$ and $b \prec c$, then $a \prec U(c)$.

Given a set $V$ with an extensional relation $\prec$, a union structure is a proto-union structure where additionally

• for all elements $a \in V$ and $c \in V$, if $a \prec U(c)$, then there exists $b \in V$ such that $a \prec b$ and $b \prec c$.

Uniqueness of $U(c)$ follows from $\prec$ being an extensional relation.

## Foundational concerns

In any material set theory, instead of postulating the mere existence of a set $U$ in which $a \in b$ and $b \in c$ implies that $a \in U$, one could add a primitive unary operation $U(c)$ which takes material sets $c$ and returns a material set $U(c)$ such that for all $a$ and $b$, $a \in b$ and $b \in c$ implies that $a \in U(c)$.