A note on guarded theories (Q1813969)

Let \(T\) be an algebraic theory in the sense of Lawvere (i.e., the objects of \(T\) are non-negative integers and the object \(n\) is a coproduct of \(n\) copies of 1). A guard family in \(T\) is a collection of morphisms \(1\to 2\) containing the canonical injections and satisfying some additional conditions. The author divides algebraic theories into three classes according to the properties of their maximal guarded families and shows by instructive examples that each class is non-empty.