Sproperties

Sproperties is a term used in theoretical disciplines to denote a class of properties of objects that are defined relative to a fixed operator S. A property P is called an S-property if it is preserved by S, in the sense that for every x in the domain X, P(x) holds if and only if P(S(x)) holds. Equivalently, P is invariant under S: P(x) = P(S(x)) for all x. S-properties form a subset of all properties on X and are often studied to understand how transformations affect truth values of predicates.

Let X be a set or structure and S: X -> X a transformation. If P and Q are S-properties, then P ∧ Q is also an S-property, as is P ∨ Q, and ¬P, under standard logical definitions, since all combinations preserve the invariance P(x) = P(S(x)). S-properties are central in areas such as formal verification, model checking, and symmetry analysis, where one seeks predicates that remain valid under program steps, state transitions, or structural symmetries.

Example: In software verification, if S represents a state transition function of a program and P states that a safety condition holds, then P is an S-property when the condition, once true, remains true after any single transition. In mathematics, P might denote a predicate describing a feature invariant under a symmetry S, such as a geometric property preserved by a reflection or rotation.

See also invariants, invariance, symmetry, and property lattices.