Based on th notion of full transformations with fixed set, in this paper, we present a novel concept of n-ary F ix(I n , Y )-full terms.This term can be considered as a generalization of strongly full terms, permutational full terms and full terms.Together with the superposition operation, one can form a Menger algebra of rank n.The freeness of such algebra with respect to a variety of algebras of the same types is discussed.Furthermore, we apply hypersubstitution theory to define a F ix(I n , Y )-full closed identity, a F ix(I n , Y )-full closed variety and present some concrete examples.

A planar semimodular lattice K is slim if M 3 is not a sublattice of K.In a recent paper, G. Czédli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the No Child Property: Let P be the ordered set of join-irreducible congruences of K. Let x, y, z ∈ P and let z be a maximal element of P. If x = y and x, y ≺ z in P, then there is no element u of P such that u ≺ x, y in P.The Swing Lemma and a standardized diagram type are used to give direct proofs of Czédli's four properties.

In this study, we give some fundamental set-theoretical solutions of Yang-Baxter equation in triangle algebras and state triangle algebras.We prove that the necessary and sufficient condition for certain mappings to be settheoretical solutions of Yang-Baxter equation on these structures is that these structures must be also MTL-(state) triangle algebras, BL-(state) triangle algebras or RL-(state) triangle algebras.In accordance with these, we recursively introduce new operators N and M.Then, we define the notion of formula on triangle algebra as a classical logic structure.Moreover, we state the relationship of transferring of set-theoretical solutions of Yang-Baxter equation among (MTL,BL, RL)-(state) triangle algebras and state (MTL,BL, RL)-(state) triangle algebras.Then, we give a scheme to explain clearly these relations.

In this paper, we define quasi-primary ideals in commutative semirings S with 1 = 0 which is a generalization of primary ideals. A proper ideal I of a semiring S is said to be a quasi-primary ideal of We also introduce the concept of 2-absoring quasi-primary ideal of a semiring S which is a generalization of quasi-primary ideal of S. A proper ideal I of a semiring S is said to be a 2-absorbing quasi-primary ideal if abc ∈ √ I implies ab ∈ √ I or bc ∈ √ I or ac ∈ √ I. Some basic results related to 2-absorbing quasi-primary ideal have also been given.

The concept of σ-filters is introduced in commutative BE-algebras and some properties of these classes of filters are studied. Some equivalent conditions are derived for every filter of a commutative BE-algebra to become a σ-filter. Some necessary and sufficient conditions are given for every regular filter of a commutative BE-algebra to become a σ-filter. A set of equivalent conditions is given for the class of all σ-filters of a commutative BE-algebra to become a sublattice to the lattice of all filters.

We generalize the concept of a fuzzy distributive lattice by introducing the concepts of a fuzzy join-distributive pair and a fuzzy join-semidistributive pair in a fuzzy lattice. A relationship among a fuzzy join-distributive pair, a fuzzy join-semidistributive pair and a fuzzy join-modular pair is proved. It is shown that for a pair of fuzzy atoms, the notions of a fuzzy join-distributive pair and a fuzzy join-semidistributive pair coincide.