Interior and closure operators on bounded residuated lattice ordered monoids

summary:$GMV$-algebras endowed with additive closure operators or with its duals-multiplicative interior operators (closure or interior $GMV$-algebras) were introduced as a non-commutative generalization of topological Boolean algebras. In the paper, the multiplicative interior and additive closure operators on $DRl$-monoids are introduced as natural generalizations of the multiplicative interior and additive closure operators on $GMV$-algebras

Classes of filters in generalizations of commutative fuzzy structures

summary:Bounded commutative residuated lattice ordered monoids ($R\ell$-monoids) are a common generalization of $\mathit {BL}$-algebras and Heyting algebras, i.e. algebras of basic fuzzy logic and intuitionistic logic, respectively. In the paper we develop the theory of filters of bounded commutative $R\ell$-monoids

Some properties of state filters in state residuated lattices

summary:We consider properties of state filters of state residuated lattices and prove that for every state filter $F$ of a state residuated lattice $X$: \begin {itemize} \item [(1)] $F$ is obstinate $\Leftrightarrow$ $L/F \cong \{0,1\}$; \item [(2)] $F$ is primary $\Leftrightarrow$ $L/F$ is a state local residuated lattice; \end {itemize} and that every g-state residuated lattice $X$ is a subdirect product of $\{X/P_{\lambda } \}$, where $P_{\lambda }$ is a prime state filter of $X$. \endgraf Moreover, we show that the quotient MTL-algebra $X/P$ of a state residuated lattice $X$ by a state prime filter $P$ is not always totally ordered, although the quotient MTL-algebra by a prime filter is totally ordered

Direct product decompositions of bounded commutative residuated ℓ\ell-monoids

summary:The notion of bounded commutative residuated $\ell$-monoid ($BCR$ $\ell$-monoid, in short) generalizes both the notions of $MV$-algebra and of $BL$-algebra. Let $\c A$ be a $BCR$ $\ell$-monoid; we denote by $\ell (\c A)$ the underlying lattice of $\c A$. In the present paper we show that each direct product decomposition of $\ell (\c A)$ determines a direct product decomposition of $\c A$. This yields that any two direct product decompositions of $\c A$ have isomorphic refinements. We consider also the relations between direct product decompositions of $\c A$ and states on $\c A$