Initial algebra semantics

Remarks

Since $h_A$ is a homomorphism, $[\![f(t_1, \dots, t_n)]\!]_A = f^A([\![t_1]\!]_A, \dots, [\![t_n]\!]_A)$, so meanings are compositional. The initial algebra of the algebras satisfying $E$ exists: it is the quotient of $T_\Sigma$ by the least congruence that contains all ground instances of the equations of $E$.

Example: let $\Sigma = \set{0, s, +}$ with arities $0, 1, 2$, and let $E$ consist of $x + 0 = x$ and $x + s(y) = s(x + y)$. The natural numbers $\N$ with zero, successor and addition satisfy $E$. Every ground term equals some $s^n(0)$ by $E$, and distinct numerals $s^m(0) \ne s^n(0)$ are separated in $\N$, so $\N$ is initial among the models of $E$.

Non-example: $\N$ is not initial in the class of all $\Sigma$-algebras. In $T_\Sigma$ the terms $s(0)$ and $s(0) + 0$ are different, but in $\N$ they have the same value $1$. A homomorphism $\N \to T_\Sigma$ would send $1$ to both terms, so there is none.

The general notion of an initial algebra for a functor contains this one: $T_\Sigma$ is the initial algebra of the polynomial functor of $\Sigma$.