Residuated lattice of L-fuzzy ideals of a ring

In 1988, given a complete Brouwerian lattice L := (L;.,.; 0, 1) and a ring A := ( A; +, center dot; -; 0) with unity 1, Swamy and Swamy (JMath Anal Appl 134:94-103, 1988) built a lattice structure, on the set of L-fuzzy ideals of A, and investigated some of its arithmetic properties. Since the residuation theory is richer than the lattice theory [see, Ciungu (Non-commutative multiple-valued logic algebras, Springer monographs in mathematics, Springer, Berlin, 2014), Galatos et al. (An algebraic glimpse at substructural logics, volume 151 of studies in logic and the foundations of mathematics, Elsevier, Amsterdam, 2007), Jipsen and Tsinakis (in: Martinez (ed) Ordered algebraic structures, Kluwer Academic Publisher, Dordrecht, 2002), Piciu (Algebras of fuzzy logic, Editura Universitaria Craiova, Craiova, 2007)], in this paper, we consider the notion of fuzzy ideals rather under a complete Brouwerian residuated lattice L := ( L;.,., , , ; 0, 1). A residuated lattice Fid(A, L) := Fid(A, L);., +,., ., ;.0, 1 is built on the set Fid(A, L) of L-fuzzy ideals of A and it is shown that the latter is both an extension of L and the residuated lattice Id(A) := Id(A); n, +, ,., ; {0}, A on the set Id(A) of ideals of A.