Equaties

Equaties are a theoretical construct in mathematics used to represent systems of equality constraints among terms of a formal language. An equatie is given by a finite set of equations t1 = s1, ..., tn = sn, where each ti and si is a term formed from operation symbols of a signature and variables. A model M for the signature satisfies the equatie if, for every variable assignment in M, all corresponding equalities hold when interpreting the terms.

Equaties generalize single equations by considering multiple equalities simultaneously. They are central to equational logic and universal algebra, and they induce equational theories: sets of sentences closed under logical consequence in the equational fragment. Basic operations on equaties include conjunction (combining sets of equalities), substitution of variables, and homomorphic images, leading to construction of quotient structures by the congruence generated by the equalities.

Examples: { x = y } specifies that x and y denote the same element; { f(x) = g(y), h(z) = w } expresses a small system of constraints. In polynomial settings, equaties can be polynomial equalities such as x^2 + y^2 = z^2.

Applications appear in constraint solving, type inference, and algebraic geometry, where solving equaties corresponds to finding points on an algebraic variety satisfying the given equalities. See also equations, equalities, constraint satisfaction, model theory, and algebraic geometry.