All General Solutions of Prešić’s Equation

S. Prešić ([6]) determined the all reproductive general solutions of the equation r(x) = 1, where r was a unary relation of a given set T . In this paper we determine the all general solutions of this equation. S. Prešić ([6]) introduced the most general concept of equations. Namely, he considered the equation of the form r(x) = 1, where r was a unary relation of a non–empty set S. He gave the formula of the all reproductive general solutions, supposing that a general solution of this equation was known. This equation was also considered by Božić ([3]), Rudeanu ([8]), Banković ([1]) and Chvalinna ([4]). S. Prešić ([5]) researched the finite equations i.e. the equations over a finite set. In the paper [7] he introduced the finite equation, the coefficients of which are from the set {0, 1} (one can prove that every finite equation is equivalent to an equation of such type) and he described the all reproductive general solutions of this equation, if particular solutions were given. The descriptions of the all general solutions and all reproductive general solutions of this equations were obtained by Banković ([2]). Rudeanu ([9]) provided simpler characterizations of the general and reproductive general solutions. Following Rudeanu’s idea from [9], we generalize, in this paper, Prešić’s result on reproductive solutions ([6]), i.e. we describe the all general solutions of the equation r(x) = 1, when a general solution of r(x) = 1 is given. Let r be a unary relation of a non–empty set T , i.e. a mapping r : T → {0, 1}. Denote by S the set of solutions of the equation r(x) = 1. Received November 5, 2001. 2000 Mathematics Subject Classification. Primary 04A05.