Free Order Sorted Universal Algebra 1 Josef Urban

(1) Let S be an order sorted signature, U0 be a strict non-empty order sorted algebra of S, and A be a subset of U0. Then A is an order sorted generator set of U0 if and only if for every OSSubset O of U0 such that O = OSClA holds OSGenO = U0. Let us consider S, let U0 be a monotone order sorted algebra of S, and let I1 be an order sorted generator set of U0. We say that I1 is osfree if and only if the condition (Def. 2) is satisfied. (Def. 2) Let U1 be a monotone non-empty order sorted algebra of S and f be a many sorted function from I1 into the sorts of U1. Then there exists a many sorted function h from U0 into U1 such that h is a homomorphism of U0 into U1 and order-sorted and h I1 = f . Let S be an order sorted signature and let I1 be a monotone order sorted algebra of S. We say that I1 is osfree if and only if: (Def. 3) There exists an order sorted generator set of I1 which is osfree. 1This work was done during author’s research visit in Bialystok, funded by the CALCULEMUS grant HPRN-CT-2000-00102.