Int. J. Reasoning-based Intelligent Systems, Vol. 5, No. 2, 2013 127 Copyright © 2013 Inderscience Enterprises Ltd. Curry algebras and propositional algebra C 1 Jair Minoro Abe Graduate Program in Production Engineering, ICET – Paulista University, R. Dr. Bacelar, 1212, CEP 04026-002, São Paulo, SP, Brazil and Institute for Advanced Studies, University of São Paulo, Rua Praça do Relógio, 109, Bloco K, 5º andar, Cidade Universitária, Caixa Postal 72.012, CEP 05508-970, São Paulo, SP, Brazil E-mail: jairabe@uol.com.br Abstract: In this expository paper, we illustrate some applications of the notion of Curry algebras, and its relationship with the concept of propositional algebra C 1 . Curry algebra was formerly introduced as a concept to study algebraic version of some non-classical systems, such as algebraic version of Da Costa’s systems C n . Keywords: Curry algebra; algebraic logic; paraconsistent logic; propositional algebra; computability; contructibility. Reference to this paper should be made as follows: Abe, J.M. (2013) ‘Curry algebras and propositional algebra C 1 ’, Int. J. Reasoning-based Intelligent Systems, Vol. 5, No. 2, pp.127–132. Biographical notes: J.M. Abe received his Doctoral degree from University of São Paulo in 1992. He is currently Full Professor at the Graduate Program in Production Engineering, Paulista University and Coordinator of the Logic and Science Theory Area of Institute for Advanced Studies, University of São Paulo, São Paulo, Brazil. His current research interest includes paraconsistent annotated logic and application in AI, automation and robotics, biomedicine and production engineering. 1 Introduction The idea of Curry system was introduced motivated by algebraic studies of some non-classical logics notably the systems C n (1 ≤ n ≤ ω) of Da Costa (1974). On the other hand, Rosenbloom (1950) presents the concept of propositional algebra, which it is close to the notion of pre-algebra studied in Curry systems. In this paper, we show that the relationship between these concepts. Some investigations made later actually have showed that all mathematical treatment of logical notions can be viewed as Curry systems. More than this, enriching or modifying the concept of Curry system, we can obtain as particular cases, the notion of logical matrix, Kripke structures, and theory of models, which are not directly coped with the problem of algebraisation. In a certain sense, we can say that the logic reduces to the study of Curry systems (Barros et al., 1995). This paper illustrates the notion of Curry system and discusses some properties and applications. It can be viewed as a kind a short overview of the matter; however, it is addressed to point out how some pre-algebraic structures can be useful. 2 Background We begin with some basic concepts. For a detailed account see Barros et al. (1995). Definition 2.1: Suppose that in a non-empty set A is fixed an equivalence relation ≡. We say that a n-ary operator ϕ on A is i-compatible with ≡ if for any x 1 , …, x i–1 , a, b, x i+1 , …, x n ∈ A, if a ≡ b, implies ϕ(x 1 , …, x i–1 , a, x i+1 , …, x n ) ≡ ϕ(x 1 , …, x i–1 , b, x i+1 , …, x n ). The operator is said to be compatible (or monotonic) with ≡ if ϕ is i-compatible with ≡ for all i = 1, …, n. A relation R on A is said to be compatible with ≡ if (x 1 , …, x n ) ∈ R and x i ≡ x’ i , i = 1, …, n then (x’ 1 , …, x’ n ) ∈ R. Definition 2.2: A Curry system is a structure < A, (≡) i∈I , (S) j∈J , (R) k∈K , (ϕ) l∈L , (C) m∈M > such that: (1) A ≠ ∅ (2) (≡) i∈I is a collection of equivalence relations (3) (S) j∈J is a family of subsets of A (4) (R) k∈K is a finite collection of relations on A