Department of Mathematics

An infinite algebra \( \mathbf{A}:=\langle A;\,F\rangle \) consists of an infinite set \( A \) and a set \( F \) of operations of finite arity on the set \( A \); it is mono-unary if \( F \) consists of a single unary operation \(f:A\rightarrow A \) and is denoted by \( \mathbf{A}^f \).

Let \( \varphi(x) \) be a formula in the first order language for \( \mathbf{A} \) with a constant symbol \( \mathbf{c}_a \) for every \( a\in A \); in one free variable \( x \).

The solution set of \( \varphi(x) \) is \( \mbox{Sol}(\varphi(x)):=\lbrace a\in A\,\mid\, \varphi(\mathbf{c}_a)\mbox{ is true in }\mathbf{A}\rbrace \subseteq A\,. \) \( \mathbf{A} \) is minimal if \( \mbox{Sol}(\varphi(x)) \) is always either finite or cofinite.

I will be talking about which \( \mathbf{A}^f \) are minimal. For instance, let \( \mathbb{N} \) and let \( \mathbb{Q} \) be the rationals; and consider the doubling function \( f(x)=2x \) on these sets; then \( \langle \mathbb{Q};\,2x\rangle \) is minimal; while \( \langle \mathbb{N};\,2x\rangle \) is not minimal.