stylistic:tone
Free universal algebras

Let S {\displaystyle S} be a set and A {\displaystyle A} be an algebraic structure of type ρ {\displaystyle \rho } generated by S {\displaystyle S}. The underlying set of this algebraic structure A {\displaystyle A}, often called its universe, is denoted by A {\displaystyle A}. Let ψ: S → A {\displaystyle \psi:S\to A} be a function. We say that (A, ψ) {\displaystyle (A,\psi)} (or informally just A {\displaystyle A}) is a free algebra of type ρ {\displaystyle \rho } on the set S {\displaystyle S} of free generators if the following universal property holds:

For every algebra B {\displaystyle B} of type ρ {\displaystyle \rho } and every function τ: S → B {\displaystyle \tau:S\to B}, where B {\displaystyle B} is the universe of B {\displaystyle B}, there exists a unique homomorphism σ: A → B {\displaystyle \sigma:A\to B} such that the following diagram commutes:

{\displaystyle }

This means that σ ∘ ψ = τ {\displaystyle \sigma \circ \psi =\tau }.