We extend the lattice embedding of the axiomatic extensions of the positive fragment of intuitionistic logic into the axiomatic extensions of intuitionistic logic to the setting of substructural logics. Our approach is algebraic and uses residuated lattices, the algebraic models for substructural logics. We generalize the notion of the ordinal sum of two residuated lattices and use it to obtain...

In this paper, the notions of fuzzy soft subalgebra and fuzzy soft convex subalgebra of a residuated lattice are introduced and some related properties are investigated. Then, we define fuzzy soft congruence on a residuated lattice and obtain the relation between fuzzy soft convex subalgebras and fuzzy soft congruence relations on residuated lattices. The concept of soft homomorphism is defined...

It is known that classical logic CL is the single maximal consistent logic over intuitionistic logic Int, which is moreover the single one even over the substructural logic FLew. On the other hand, if we consider maximal consistent logics over a weaker logic, there may be uncountablymany of them. Since the subvariety lattice of a given variety V of residuated lattices is dually isomorphic to th...

We introduce a new product bilattice construction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize ...

An interesting generalization of hoop-algebras and commutative residuated lattices is the concept quasi-ordered systems (shortly QRS) introduced in 2018 by Bonzio Chajda. Quasi-ordered system an integral monoid with two internal binary operations interconnected a residuation connection. This specificity reason for complexity this algebraic structure existence significant number substructures it...

Infinite fields are not finitely generated rings. Similar question is considered for further algebraic structures, mainly commutative semirings. In this case, purely algebraic methods fail and topological properties of integral lattice points turn out to be useful. We prove that a commutative semiring that is a group with respect to multiplication, can be two-generated only if it belongs to the...

Let L be a complete residuated lattice. Then we show that any L-preorder can be represented both by an implication-based graded inclusion as defined [1] and by a similarity-based graded inclusion as defined in [2]. Also, in accordance with a duality between [0,1]-orders and quasi-metrics, we obtain two corresponding representation theorems for quasi-metrics.

We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pa...

The assertional logic S(BCIA) of the quasivariety of BCI-algebras (in Iseki's sense) is axiomatized, relative to pure implicational logic BCI, by the rule x, y, x → y (G) (see [1]). Alternatively, the role of (G) can be played by x x → (y → y) (1) (see [2]). The formula (x → x) → (y → y) (2) is a theorem of S(BCIA). In [2, Proposition 22] we claimed erroneously that, relative to BCI, the axiom ...

This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones only evaluated in 0 and 1. ...