Hahn’s famous structure theorem states that totally ordered Abelian groups can be embedded in the lexicographic product of real groups. Our main theorem extends this structural description to order-dense, commutative, group-like residuated chains, which has only finitely many idempotents. It is achieved via the so-called partial-lexicographic product construction (to be introduced here) using t...

Bipolar max-? equations, with ? a triangular norm, have recently become popular research topic embedded in the broad field of fuzzy relational equations. In this paper, we lift work from restrictive setting real unit interval — obfuscating underlying lattice-theoretical essence to general complete distributive symmetric residuated lattices, allowing build upon vast body knowledge on unipolar su...

EQ-algebras were introduced by Novak (2006) as an algebraic structure of truth values for fuzzy-type theory (FFT). Novák and De Baets (2009) various kinds such good, residuated, IEQ-algebras. In this paper, we define the notion (pre)ideal in bounded (BEQ-algebras) investigate some properties. Then, introduce a congruence relation on good BEQ-algebras using ideals, then, solve open problem Paad ...

We give the simple general principle of studying the relations among fuzzy t-filters on residuated lattices. Using the general principle, we can easily determine the relations among fuzzy t-filters on different logical algebras.

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The Bandler–Kohout Subproduct (BKS) inference mechanism is one of the two established fuzzy relational inference (FRI) mechanisms, the other being Zadeh’s Compositional Rule of Inference (CRI). Both these FRIs are known to possess many desirable properties. It can be seen that many of these desirable properties are due to the rich underlying structure, viz., the residuated algebra, from which t...

On the real unit interval, the notion of a Girard monoid coincides with the notion of a t-norm-based residuated lattice with strong induced negation. A geometrical approach toward these Girard monoids, based on the notion of rotation invariance, is turned in an adequate axiomatization for the Involutive Monoidal T-norm-based residuated Logic (IMTL).

in this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. furthermore,...

Lattice-ordered monoids are important backgrounds and algebraic foundations of residuum in general. The t-norm based lattices are investigated widely in fuzzy models, but in recent time while researching new approximate reasoning methods in soft computing based models and fuzzy models, the investigations are focused on new types of operators, like uninorms. It is necessary to find and define co...

In this paper, we introduce the notions of L-fuzzy quasi-uniformities and Lfuzzy interior operators in complete residuated lattices. We investigate the L-fuzzy quasiuniformities induced by L-fuzzy interior operators. We study the relationships between Lfuzzy interior operators and L-fuzzy quasi-uniformities. We give their examples. AMS Subject Classification: 03E72, 06A15, 06F07, 54F05