Order Sorted Algebras 1 Josef Urban

Let I be a set, let f be a many sorted set indexed by I, and let p be a finite sequence of elements of I. Observe that f · p is finite sequence-like. Let S be a non empty many sorted signature. A sort symbol of S is an element of S. Let S be a non empty many sorted signature. An operation symbol of S is an element of the operation symbols of S. Let S be a non void non empty many sorted signature and let o be an operation symbol of S. Then the result sort of o is an element of S. We consider overloaded many sorted signatures as extensions of many sorted signature as systems 〈 a carrier, operation symbols, an overloading, an arity, a result sort 〉, where the carrier is a set, the operation symbols constitute a set, the overloading is an equivalence relation of the operation symbols, the arity is a function from the operation symbols into the carrier∗, and the result sort is a function from the operation symbols into the carrier. We consider relation sorted signatures as extensions of many sorted signature and relational structure as systems 〈 a carrier, an internal relation, operation symbols, an arity, a result sort 〉, where the carrier is a set, the internal relation is a binary relation on the carrier, the operation symbols constitute a set, the arity is a function from the operation symbols into the carrier∗, and the result sort is a function from the operation symbols into the carrier. We consider overloaded relation sorted signatures as extensions of overloaded many sorted signature and relation sorted signature as systems 〈 a carrier, an internal relation, operation symbols, an overloading, an arity, a result sort 〉, where the carrier is a set, the internal relation is a binary relation on the carrier, the operation symbols constitute a set, the overloading is an equivalence relation of the operation symbols, the arity is a function from the operation symbols into the carrier∗, and the result sort is a function from the operation symbols into the carrier.