Dually residuated lattice-ordered monoids (DRl-monoids) generalize lattice-ordered groups and include also some algebras related to fuzzy logic (e.g. GMV-algebras and pseudo BL-algebras). In the paper, we give some necessary and sufficient conditions for a DRl-monoid to be representable (i.e. a subdirect product of totally ordered DRl-monoids) and we prove that the class of representable DRl-monoids is a variety.
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 deal with unbounded dually residuated lattices that generalize pseudo -algebras in such a way that every principal order-ideal is a pseudo -algebra. We describe the connections of these generalized pseudo -algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo -algebra by means of the positive cone of a suitable -group . We prove that the lattice of all (normal) ideals of and the lattice of all (normal) convex -subgroups of are isomorphic....
In this note we describe the structure of dually residuated -monoids (-monoids) that have no non-trivial convex subalgebras.
Lattice-ordered groups, as well as -algebras (pseudo -algebras), are both particular cases of dually residuated lattice-ordered monoids (-monoids for short). In the paper we study ideals of lower-bounded -monoids including -algebras. Especially, we deal with the connections between ideals of a -monoid and ideals of the lattice reduct of .
It is shown that pseudo -algebras are categorically equivalent to certain bounded -monoids. Using this result, we obtain some properties of pseudo -algebras, in particular, we can characterize congruence kernels by means of normal filters. Further, we deal with representable pseudo -algebras and, in conclusion, we prove that they form a variety.
Dually residuated lattice-ordered monoids (-monoids for short) generalize lattice-ordered groups and include for instance also -algebras (pseudo -algebras), a non-commutative extension of -algebras. In the present paper, the spectral topology of proper prime ideals is introduced and studied.
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 . 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...
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...
In the paper we deal with weak Boolean products of bounded dually residuated -monoids (DR-monoids). Since bounded DRl-monoids are a generalization of pseudo MV-algebras and pseudo BL-algebras, the results can be immediately applied to these algebras.
We consider algebras determined by all normal identities of -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 -lattice, and another one based on a normalization of a lattice-ordered group.
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