| Title:
|
MV-algebras are categorically equivalent to a class of $\scr{DR}l\sb {1(i)}$-semigroups (English) |
| Author:
|
Rachůnek, Jiří |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
123 |
| Issue:
|
4 |
| Year:
|
1998 |
| Pages:
|
437-441 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the paper it is proved that the category of \MV-algebras is equivalent to the category of bounded \DRl-semigroups satisfying the identity $1-(1-x)=x$. Consequently, by a result of D. Mundici, both categories are equivalent to the category of bounded commutative \BCK-algebras. (English) |
| Keyword:
|
categorical equivalence |
| Keyword:
|
bounded \BCK-algebra |
| Keyword:
|
\MV-algebra |
| Keyword:
|
\DRl-semigroup |
| MSC:
|
03G20 |
| MSC:
|
06D30 |
| MSC:
|
06D35 |
| MSC:
|
06F05 |
| MSC:
|
06F35 |
| idZBL:
|
Zbl 0934.06014 |
| idMR:
|
MR1667115 |
| DOI:
|
10.21136/MB.1998.125964 |
| . |
| Date available:
|
2009-09-24T21:34:04Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/125964 |
| . |
| Reference:
|
[1] C. C. Chang: Algebraic analysis of many valued logics.Trans. Amer. Math. Soc. 88 (1958), 467-490. Zbl 0084.00704, MR 0094302, 10.1090/S0002-9947-1958-0094302-9 |
| Reference:
|
[2] C. C. Chang: A new proof of the completeness of the Lukasiewicz axioms.Trans. Amer. Math. Soc. 93 (1959), 74-80. Zbl 0093.01104, MR 0122718 |
| Reference:
|
[3] R. Cignoli: Free lattice-ordered abelian groups and varieties of MV-algebras.Proc. IX. Latin. Amer. Symp. Math. Logic, Part 1, Not. Log. Mat. 38 (1993), 113-118. Zbl 0827.06012, MR 1332526 |
| Reference:
|
[4] K. Iséki, S. Tanaka: An introduction to the theory of BCK-algebras.Math. Japonica. 23 (1978), 1-26. MR 0500283 |
| Reference:
|
[5] T. Kovář: A general theory of dually residuated lattice ordered monoids.Thesis, Palacky Univ. Olomouc, 1996. |
| Reference:
|
[6] T. Kovář: Two remarks on dually residuated lattice ordered semigroups.Math. Slovaca. To appear. MR 1804468 |
| Reference:
|
[7] F. Lacava: Some properties of L-algebras and existencially closed L-algebras.Boll. Un. Mat. Ital., A(5) 16 (1979), 360-366. (In Italian.) MR 0541775 |
| Reference:
|
[8] D. Mundici: Interpretation of AF C*-algebras in Lukasiewicz sentential calculus.J. Funct. Analys. 65 (1986), 15-63. MR 0819173, 10.1016/0022-1236(86)90015-7 |
| Reference:
|
[9] D. Mundici: MV-algebras are categorically equivalent to bounded commutative BCK-algebras.Math. Japonica 31 (1986), 889-894. Zbl 0633.03066, MR 0870978 |
| Reference:
|
[10] J. Rachůnek: DRI-semigroups and MV-algebras.Czechoslovak Math. J. 123 (1998), 365-372. MR 1624268, 10.1023/A:1022801907138 |
| Reference:
|
[11] K. L. N. Swamy: Dually residuated lattice ordered semigroups.Math. Ann. 159 (1965), 105-114. Zbl 0138.02104, MR 0183797, 10.1007/BF01360284 |
| Reference:
|
[12] S. Tanaka: On $\Lambda$-commutative algebras.Math. Sem. Notes Kobe 3 (1975), 59-64. Zbl 0324.02053, MR 0419222 |
| Reference:
|
[13] T. Traczyk: On the variety of bounded commutative BCK-algebras.Math. Japonica 24 (1979), 283-292. Zbl 0422.03038, MR 0550212 |
| Reference:
|
[14] H. Yutani: Quasi-commutative BCK-algebras and congruence relations.Math. Sem. Notes Kobe 5 (1977), 469-480. Zbl 0375.02053, MR 0498112 |
| . |
