In first-order logic, a structure is a nonempty set with operations and relations (called the basic operations and relations of the structure). That is how Tarski defined structures. He could have defined structures differently; there were a number of reasonable options. For our purposes here, a variant of Tarski's notion is more appropriate. Respecting tradition, we do not redefine structures. Rather, we modify the notion of structure and give the new notion a new name.
Structures without relations are called algebras in the branch of mathematics called universal algebra. Restrict attention to algebras with distinct nullary operations true and false and define basic relations as basic operations taking only the Boolean values true and false. Further restrict attention to algebras with the equality relation and the usual Boolean operations. (We will specify later the values of the Boolean operations outside their natural domains.) The resulting notion of algebra is our variant of the notion of structure with equality. It allows us to write quantifier-free formulas as terms.
Actually, we are interested in multi-sorted structures with partial
operations. The sorts can be given by unary relations (they will be called
universes and the whole underlying set of a structure will be called the
superuniverse). To deal with partial functions, further restrict attention
to algebras with a nullary operation undef, different from true and
false, and interpret an operation
f as undefined at a tuple if f(
) = undef.
These algebras will be called static
algebras or states. Their operations will be called functions.
In the following subsections, we start anew and define static algebras from scratch, establishing terminology on the way.