We deal with unbounded dually residuated lattices that generalize pseudo $MV$-algebras in such a way that every principal order-ideal is a pseudo $MV$-algebra. We describe the connections of these generalized pseudo $MV$-algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo $MV$-algebra $A$ by means of the positive cone of a suitable $\ell $-group $G_A$. We prove that the lattice of all (normal) ideals of $A$ and the lattice of all (normal) convex $\ell $-subgroups of $G_A$ are isomorphic. We also introduce the concept of Archimedeanness and show that every Archimedean generalized pseudo $MV$-algebra is commutative.
In this paper, we introduce the concept of an ideal of a noncommutative dually residuated lattice ordered monoid and we show that congruence relations and certain ideals are in a one-to-one correspondence.
We generalize the concept of an integral residuated lattice to join-semilattices with an upper bound where every principal order-filter (section) is a residuated semilattice; such a structure is called a {\it sectionally residuated semilattice}. Natural examples come from propositional logic. For instance, implication algebras (also known as Tarski algebras), which are the algebraic models of the implication fragment of the classical logic, are sectionally residuated semilattices such that every section is even a Boolean algebra. A similar situation rises in case of the Łukasiewicz multiple-valued logic where sections are bounded commutative BCK-algebras, hence MV-algebras. Likewise, every integral residuated (semi)lattice is sectionally residuated in a natural way. We show that sectionally residuated semilattices can be axiomatized as algebras $(A,r,\rightarrow ,\rightsquigarrow,1)$ of type $\langle 3,2,2,0\rangle $ where $(A,\rightarrow, \rightsquigarrow,1)$ is a $\{\rightarrow ,\rightsquigarrow,
1\}$-subreduct of an integral residuated lattice. We prove that every sectionally residuated {\it lattice} can be isomorphically embedded into a residuated lattice in which the ternary operation $r$ is given by $r(x,y,z)=(x\cdot y)ěe z$. Finally, we describe mutual connections between involutive sectionally residuated semilattices and certain biresiduation algebras.
We consider algebras determined by all normal identities of MV -algebras, i.e. algebras of many-valued logics. For such algebras, we present a representation based on a normalization of a sectionally involutioned lattice, i.e. a q-lattice, and another one based on a normalization of a lattice-ordered group.
It is shown that pseudo BL-algebras are categorically equivalent to certain bounded DRl-monoids. Using this result, we obtain some properties of pseudo BL-algebras, in particular, we can characterize congruence kernels by means of normal filters. Further, we deal with representable pseudo BL-algebras and, in conclusion, we prove that they form a variety.
Dually residuated lattice-ordered monoids (DRl-monoids for short) generalize lattice-ordered groups and include for instance also GMV -algebras (pseudo MV -algebras), a non-commutative extension of MV -algebras. In the present paper, the spectral topology of proper prime ideals is introduced and studied.
Sectionally pseudocomplemented semilattices are an extension of relatively pseudocomplemented semilattices—they are meet-semilattices with a greatest element such that every section, i.e., every principal filter, is a pseudocomplemented semilattice. In the paper, we give a simple equational characterization of sectionally pseudocomplemented semilattices and then investigate mainly their congruence kernels which leads to a characterization of subdirectly irreducible sectionally pseudocomplemented semilattices.
In the paper we deal with weak Boolean products of bounded dually residuated l-monoids (DRl-monoids). Since bounded DRl-monoids are a generalization of pseudo MValgebras and pseudo BL-algebras, the results can be immediately applied to these algebras.