In this expository paper we present an introduction to the Curry Systems. Over the past decades many non-classical logics have appeared and they have inspired some algebraic structures underlying such systems. In this paper, we discuss some applications of the concept of Curry algebra for algebraization of some paraconsistent, paracomplete, and non-alethic logics. Such concept is also correlated with some fundamental themes in logic, such as computability, constructibility, topology, and many other basic branches.

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A Hales-Jewett set is a set of words of a given length on a specified alphabet with the property that whenever it is 2-colored, there must be a monochromatic combinatorial line. We show that any Hales-Jewett set consisting of length 4 words on the alphabet {1, 2, 3} must have at least 25 members and produce an example of a minimal Hales-Jewett set with 37 members.

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In this paper, we prove three fundamental theorems of triangle constructive trices. The representative of non-distributive trice is a triangle constructive trice. The first theorem is that two elements trice which composes the distributive trice doesn’t exist in a triangle constructive trice at all. The second is a theorem to compose the order reversing operation on triangle constructive trices. The third is a theorem concerning homomorphism of triangle constructive trices. The homomorphism of two operations leads the homomorphism of another operation in triangle constructive trices.

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An endomorphism of an algebra A is said to be strong if it is compatible with every congruence on A; and A is said to have the strong endomorphism kernel property if every congruence on A, other than the universal congruence, is the kernel of a strong endomorphism on A. Here we describe, by way of Priestley duality, the distributive double p-algebras that have the strong endomorphism kernel property.

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This paper discusses some properties of convex subsets of affine spaces over the principal ideal domains Z[1/p], where p is a prime number. In particular, it is shown that such convex sets are equivalent to certain algebras with finitely many operations, and a minimal number of generators for the segments of the line Z[1/p] is provided. In the case of p = 3, relations between convex sets and a certain groupoid variety are discussed.

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In this note, we shall report solutions of a number theoretic problem by Moret Blanc, which Professor Is´eki studied.

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Dyson’s Circular Orthogonal, Unitary, and Symplectic Ensembles are measure spaces modeling families of physical systems with specific spin and timereversal properties. In this paper, it is shown that each of the three ensembles carries a transitive right quasigroup structure, such that automorphisms of the measure space are automorphisms of the quasigroup structure.

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We shall review the work of Professor Kiyoshi Iseki on topology.

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This survey paper contains a brief account of the literature on preservation of continuity under various convergences from the beginning to current research. We include stronger forms of continuity such as uniform continuity, proximal continuity, strong uniform continuity and weaker forms of continuity such as quasi-continuity, near continuity and almost continuity. A few proofs of the results are given and references are given for others. A standard reference for this research area is [28]

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We survey some interesting inequalities which can be derived from convex functions.

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In this note, we make a review of the weak elementary Euclidean Pasch free geometry.

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