The aim of this paper is to give a new equivalent set of axioms for MV-algebras, and to show that the axioms are independent. In addition to this, we handle Yang-Baxter equation problem. In conclusion, we construct a new set-theoretical solution for the Yang-Baxter equation by using MV-algebras.

Abstract:In this study, a term operation Sheffer stroke is presented in a given basic algebra A and the properties of the Sheffer stroke reduct of A are examined. In addition, we qualify such Sheffer stroke basic algebras. Finally, we construct a bridge between Sheffer stroke basic algebras and Boolean algebras.

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In this paper, we analyze the algebraic properties of categorical syllogisms by constructing a logical calculus system called Syllogistic Logic with Carroll Diagrams (SLCD). We prove that any categorical syllogism is valid if and only if it is provable in this system. For this purpose, we explain firstly the quantitative relation between two terms by means of bilateral diagrams and we clarify premises via bilateral diagrams. Afterwards, we input the data taken from bilateral diagrams, on the trilateral diagram. With the help of the elimination method, we obtain a conclusion that is transformed from trilateral diagram to bilateral diagram. Subsequently, we study a syllogistic conclusion mapping which gives us a conclusion obtained from premises. Finally, we allege valid forms of syllogisms using algebraic methods, and we examine their algebraic properties, and also by using syllogisms, we construct algebraic structures, such as lattices, Boolean algebras, Boolean rings, and many-valued algebras (MV-algebras).

The main purpose of this paper is to define a Heyting algebra on categorical syllogisms. For this aim, we explain categorical syllogisms by the diagrammatic method, which gives us a suitable treatment to logical reasoning with Caroll's diagrams. In this regard, we represent the quantitative relations between syllogisms' terms by means of bilateral diagrams. Finally, we construct a system, which is a Heyting algebra, for examining categorical syllogisms by using sets.