Structures

Overview

A binary structure is a pair $⟨A,R⟩$ consisting of a set $A$ and a binary relation $R$ on $A$, i.e. $R\subseteq A\times A$.

Isomorphisms

Let $⟨A,R⟩$ and $⟨B,S⟩$ be binary structures. An isomorphism from $⟨A,R⟩$ onto $⟨B,S⟩$ is a one-to-one function $f$ from $A$ onto $B$ such that, for all $x,y\in A$, $$xRy \Leftrightarrow f(x) S f(y).$$

If such an isomorphism exists, then we say that $⟨A,R⟩$ is isomorphic to $⟨B,S⟩$, denoted $⟨A,R⟩\cong ⟨B,S⟩$.

Bibliography

• Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).