Join-semilattices whose sections are residuated po-monoids
- Title:
- Join-semilattices whose sections are residuated po-monoids
- Creator:
- Chajda, Ivan and Kühr, Jan
- Identifier:
- https://cdk.lib.cas.cz/client/handle/uuid:75632244-ec77-499f-a1f5-32960555998b
uuid:75632244-ec77-499f-a1f5-32960555998b - Subject:
- residuated lattice, residuated semilattice, biresiduation algebra, pseudo-MV-algebra, sectionally residuated semilattice, and sectionally residuated lattice
- Type:
- model:article and TEXT
- Format:
- bez média and svazek
- Description:
- 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.
- Language:
- English
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/
policy:public - Coverage:
- 1107-1127
- Source:
- Czechoslovak Mathematical Journal | 2008 Volume:58 | Number:4
- Harvested from:
- CDK
- Metadata only:
- false
The item or associated files might be "in copyright"; review the provided rights metadata:
- http://creativecommons.org/publicdomain/mark/1.0/
- policy:public