| Title:
|
Implication and equivalential reducts of basic algebras (English) |
| Author:
|
Chajda, Ivan |
| Author:
|
Kolařík, Miroslav |
| Author:
|
Švrček, Filip |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
49 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
21-36 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A term operation implication is introduced in a given basic algebra $\mathcal {A}$ and properties of the implication reduct of $\mathcal {A}$ are treated. We characterize such implication basic algebras and get congruence properties of the variety of these algebras. A term operation equivalence is introduced later and properties of this operation are described. It is shown how this operation is related with the induced partial order of $\mathcal {A}$ and, if this partial order is linear, the algebra $\mathcal {A}$ can be reconstructed by means of its equivalential reduct. (English) |
| Keyword:
|
Basic algebra |
| Keyword:
|
implication algebra |
| Keyword:
|
implication reduct |
| Keyword:
|
equivalential algebra |
| Keyword:
|
equivalential reduct |
| MSC:
|
03G25 |
| MSC:
|
06C15 |
| MSC:
|
06D35 |
| MSC:
|
08A62 |
| idZBL:
|
Zbl 1235.06010 |
| idMR:
|
MR2796944 |
| . |
| Date available:
|
2011-02-18T07:34:34Z |
| Last updated:
|
2013-09-18 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141414 |
| . |
| Reference:
|
[1] Abbott, J. C.: Orthoimplication algebras. Studia Logica 35 (1976), 173–177. Zbl 0331.02036, MR 0441794, 10.1007/BF02120879 |
| Reference:
|
[2] Chajda, I., Eigenthaler, G., Länger, H.: Congruence Classes in Universal Algebra. Heldermann Verlag, Lemgo (Germany), 2003. MR 1985832 |
| Reference:
|
[3] Chajda, I., Halaš, R., Kühr, J.: Semilattice Structures. Heldermann Verlag, Lemgo (Germany), 2007. Zbl 1117.06001, MR 2326262 |
| Reference:
|
[4] Chajda, I., Kolařík, M.: Independence of axiom system of basic algebras. Soft Computing 13, 1 (2009), 41–43. Zbl 1178.06007, 10.1007/s00500-008-0291-2 |
| Reference:
|
[5] Cignoli, R. L. O., D’Ottaviano, I. M. L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht–Boston–London, 2000. MR 1786097 |
| Reference:
|
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| Reference:
|
[7] Megill, N. D., Pavičić, M.: Quantum implication algebras. Int. J. Theor. Phys. 42, 12 (2003), 2807–2822. Zbl 1039.81007, MR 2023776, 10.1023/B:IJTP.0000006007.58191.da |
| Reference:
|
[8] Słomczynska, K.: Equivalential algebras. Part I: Representation. Algebra Universalis 35 (1996), 524–547. MR 1392281, 10.1007/BF01243593 |
| Reference:
|
[9] Tax, R. E.: On the intuitionistic equivalential calcalus. Notre Dame Journal of Formal Logic 14 (1973), 448–456. MR 0329866, 10.1305/ndjfl/1093891099 |
| . |
