| Title:
|
Join-semilattices whose sections are residuated po-monoids (English) |
| Author:
|
Chajda, Ivan |
| Author:
|
Kühr, Jan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
1107-1127 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
residuated lattice |
| Keyword:
|
residuated semilattice |
| Keyword:
|
biresiduation algebra |
| Keyword:
|
pseudo-MV-algebra |
| Keyword:
|
sectionally residuated semilattice |
| Keyword:
|
sectionally residuated lattice |
| MSC:
|
06D35 |
| MSC:
|
06F05 |
| MSC:
|
06F35 |
| idZBL:
|
Zbl 1174.06324 |
| idMR:
|
MR2471170 |
| . |
| Date available:
|
2010-07-21T08:12:21Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140444 |
| . |
| Reference:
|
[1] Abbott, J. C.: Semi-boolean algebra.Matem. Vestnik 4 (1967), 177-198. Zbl 0153.02704, MR 0239957 |
| Reference:
|
[2] Alten, C. J. van: Representable biresiduated lattices.J. Algebra 247 (2002), 672-691. MR 1877868, 10.1006/jabr.2001.9039 |
| Reference:
|
[3] Alten, C. J. van: On varieties of biresiduation algebras.Stud. Log. 83 (2006), 425-445. MR 2250119, 10.1007/s11225-006-8312-6 |
| Reference:
|
[4] Ceterchi, R.: Pseudo-Wajsberg algebras.Mult.-Valued Log. 6 (2001), 67-88. Zbl 1013.03074, MR 1817437 |
| Reference:
|
[5] Chajda, I., Halaš, R., Kühr, J.: Implication in MV-algebras.Algebra Univers. 52 (2004), 377-382. MR 2120523 |
| Reference:
|
[6] Cignoli, R. L. O., D'Ottaviano, I. M. L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning.Kluwer Acad. Publ., Dordrecht (2000). Zbl 0937.06009, MR 1786097 |
| Reference:
|
[7] Galatos, N., Tsinakis, C.: Generalized MV-algebras.J. Algebra 283 (2005), 254-291. Zbl 1063.06008, MR 2102083, 10.1016/j.jalgebra.2004.07.002 |
| Reference:
|
[8] Georgescu, G., Iorgulescu, A.: Pseudo-MV algebras.Mult.-Valued Log. 6 (2001), 95-135. Zbl 1014.06008, MR 1817439 |
| Reference:
|
[9] Georgescu, G., Iorgulescu, A.: Pseudo-BCK algebras: An extension of BCK algebras.Proc. of DMTCS'01: Combinatorics, Computability and Logic, London (2001), 97-114. Zbl 0986.06018, MR 1934824 |
| Reference:
|
[10] Jipsen, P., Tsinakis, C.: A survey of residuated lattices.Ordered Algebraic Structures (J. Martinez, ed.), Kluwer Acad. Publ., Dordrecht (2002), 19-56. Zbl 1070.06005, MR 2083033 |
| Reference:
|
[11] Kühr, J.: Pseudo BCK-algebras and residuated lattices.Contr. Gen. Algebra 16 (2005), 139-144. MR 2166954 |
| Reference:
|
[12] Kühr, J.: Commutative pseudo BCK-algebras.(to appear) in Southeast Asian Bull. Math. MR 2524913 |
| Reference:
|
[13] Leuştean, I.: Non-commutative Łukasiewicz propositional logic.Arch. Math. Log. 45 (2006), 191-213. MR 2209743, 10.1007/s00153-005-0297-8 |
| Reference:
|
[14] Rachůnek, J.: A non-commutative generalization of MV-algebras.Czech. Math. J. 52 (2002), 255-273. MR 1905434, 10.1023/A:1021766309509 |
| Reference:
|
[15] Ward, M., Dilworth, R. P.: Residuated lattices.Trans. Am. Math. Soc. 45 (1939), 335-354. Zbl 0021.10801, MR 1501995, 10.1090/S0002-9947-1939-1501995-3 |
| . |
