Using the notion of anti fuzzy points and its besideness to and non-quasi-coincidence with fuzzy set, new concepts of anti fuzzy subalgebras in BCK/BCI-algebras are introduced and their inter-relations and related properties are investigated in [3]. The notion of the new fuzzy subalgebra with thresholds is introduced and relationship between this notion and the new fuzzy subalgebra of a BCK/BCI-algebra of [3] are studied.

The Jun’s YεJ -fuzzy set is applied to the Sheffer stroke BL-algebra, and the notions of an (∈, ∈)- YεJ -fuzzy quasisubalgebra and an (∈, ∈ ∨q)-YεJ -fuzzy quasi-subalgebra in Sheffer stroke BL-algebras are introduced, and several properties are explored. Characterizations of (∈, ∈)- YεJ - fuzzy quasi-subalgebras and (∈, ∈ ∨q)- YεJ -fuzzy quasisubalgebras are investigated, and the relationships between the fuzzy quasi-subalgebra, the (∈, ∈)- YεJ -fuzzy quasi- subalgebra and the (∈, ∈ ∨q)- YεJ -fuzzy quasisubalgebra are considered. Conditions for a fuzzy set to be an (∈, ∈∨q)- YεJ -fuzzy quasi-subalgebra are displayed.

The notion of (closed) Lukasiewicz fuzzy ideal is introduced, and several properties are investigated.The relationship between Lukasiewicz fuzzy subalgebra and Lukasiewiczfuzzy ideal is discussed, and characterization of a Lukasiewicz fuzzy ideal is considered.Conditions for a Lukasiewicz fuzzy subalgebra to be a Lukasiewicz fuzzy ideal are provided, andconditions for the $\in$-set, $q$-set and $O$-set to be ideals are explored.

In this paper, we define the notions of fuzzy congruence relations and fuzzy convex sub algebras on a commutative resituated lattice and we obtain some related results. In particular, we will show that there exists a one to one correspondence between the set of all fuzzy congruence relations and the set of all fuzzy convex subalgcbras on a commutative residuatcd lattice. Then we study fuzzy convex sub algebras of an integral commutative residuated lattice and will prove that fuzzy filters and fuzzy convex suhalgebras of an integral commutative residuated lattice coincide.

In 1993, Moneyhun showed that if L is a Lie algebra such that dim(L/Z(L)) = n, then dim(L^2) 1/2n(n-1). The author and Saeedi investigated the converse of Moneyhun's result under some con-ditions. In this paper, We extend their results to obtain several upper bounds for the dimension of a Lie algebra L in terms of dimension of L2, where L^2 is the derived subalgebra. Moreover, we give an upper bound for the dimension of the c-nilpotent multiplier of a pair of Lie algebras.

Schur proved that if the center of a group G has finite index, then the derived subgroup G′ is also finite. Moneyhun proved that if L is a Lie algebra such that dim(L/Z(L)) = n, then dim(L^2) ≤1/2n(n-1) In this paper, we extend the converse of Moneyhun’s theorem. Also, we prove a wellknown result of nilpotent Lie algebras by using a different technique