Join-semilattices whose sections are residuated po-monoids

Ivan Chajda; Jan Kühr

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 1107-1127
  • ISSN: 0011-4642

Abstract

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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 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 , → , ⇝ , 1 ) of type 〈 3 , 2 , 2 , 0 〉 where ( A , → , ⇝ , 1 ) is a { → , ⇝ , 1 } -subreduct of an integral residuated lattice. We prove that every sectionally residuated lattice can be isomorphically embedded into a residuated lattice in which the ternary operation r is given by r ( x , y , z ) = ( x · y ) ě e z . Finally, we describe mutual connections between involutive sectionally residuated semilattices and certain biresiduation algebras.

How to cite

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Chajda, Ivan, and Kühr, Jan. "Join-semilattices whose sections are residuated po-monoids." Czechoslovak Mathematical Journal 58.4 (2008): 1107-1127. <http://eudml.org/doc/37890>.

@article{Chajda2008,
abstract = {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 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 $\lbrace \rightarrow ,\rightsquigarrow ,1\rbrace $-subreduct of an integral residuated lattice. We prove that every sectionally residuated 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.},
author = {Chajda, Ivan, Kühr, Jan},
journal = {Czechoslovak Mathematical Journal},
keywords = {residuated lattice; residuated semilattice; biresiduation algebra; pseudo-MV-algebra; sectionally residuated semilattice; sectionally residuated lattice; residuated lattice; residuated semilattice; biresiduation algebra; pseudo-MV-algebra; sectionally residuated semilattice; sectionally residuated lattice},
language = {eng},
number = {4},
pages = {1107-1127},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Join-semilattices whose sections are residuated po-monoids},
url = {http://eudml.org/doc/37890},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Chajda, Ivan
AU - Kühr, Jan
TI - Join-semilattices whose sections are residuated po-monoids
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 1107
EP - 1127
AB - 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 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 $\lbrace \rightarrow ,\rightsquigarrow ,1\rbrace $-subreduct of an integral residuated lattice. We prove that every sectionally residuated 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.
LA - eng
KW - residuated lattice; residuated semilattice; biresiduation algebra; pseudo-MV-algebra; sectionally residuated semilattice; sectionally residuated lattice; residuated lattice; residuated semilattice; biresiduation algebra; pseudo-MV-algebra; sectionally residuated semilattice; sectionally residuated lattice
UR - http://eudml.org/doc/37890
ER -

References

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