| Title:
|
Cardinal and ordinal arithmetics of $n$-ary relational systems and $n$-ary ordered sets (English) |
| Author:
|
Karásek, Jiří |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
123 |
| Issue:
|
3 |
| Year:
|
1998 |
| Pages:
|
249-262 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The aim of this paper is to define and study cardinal (direct) and ordinal operations of addition, multiplication, and exponentiation for $n$-ary relational systems. $n$-ary ordered sets are defined as special $n$-ary relational systems by means of properties that seem to suitably generalize reflexivity, antisymmetry, and transitivity from the case of $n=2$ or 3. The class of $n$-ary ordered sets is then closed under the cardinal and ordinal operations. (English) |
| Keyword:
|
cardinal sum |
| Keyword:
|
cardinal product |
| Keyword:
|
ordinal sum |
| Keyword:
|
ordinal product |
| Keyword:
|
$n$-ary relational system |
| Keyword:
|
$n$-ary ordered set |
| Keyword:
|
cardinal power |
| Keyword:
|
ordinal power |
| MSC:
|
03E05 |
| MSC:
|
03E10 |
| MSC:
|
04A05 |
| MSC:
|
06A99 |
| MSC:
|
08A02 |
| idZBL:
|
Zbl 0933.08001 |
| idMR:
|
MR1645434 |
| DOI:
|
10.21136/MB.1998.126071 |
| . |
| Date available:
|
2009-09-24T21:31:46Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126071 |
| . |
| Reference:
|
[1] G. Birkhoff: An extended arithmetic.Duke Math. J. 3 (1937), 311-316. Zbl 0016.38702, MR 1545989, 10.1215/S0012-7094-37-00323-5 |
| Reference:
|
[2] G. Birkhoff: Generalized arithmetic.Duke Math. J. V (1942), 283-302. Zbl 0060.12609, MR 0007031, 10.1215/S0012-7094-42-00921-9 |
| Reference:
|
[3] G. Birkhoff: Lattice Theory.Amer. Math. Soc., Providence, Rhode Island, Third Edition, 1973. MR 0227053 |
| Reference:
|
[4] M. M. Day: Arithmetic of ordered systems.Trans. Amer. Math. Soc. 58 (1945), 1-43. Zbl 0060.05813, MR 0012262, 10.1090/S0002-9947-1945-0012262-4 |
| Reference:
|
[5] V. Novák: On a power of relational structures.Czechoslovak Math. J. 35 (1985), 167-172. MR 0779345 |
| Reference:
|
[6] V. Novák M. Novotný: Binary and ternary relations.Math. Bohem. 117(1992), 283-292. MR 1184541 |
| Reference:
|
[7] V. Novák M. Novotný: Pseudodimension of relational structures.Czechoslovak Math. J. (submitted). MR 1708362 |
| Reference:
|
[8] J. Šlapal: Direct arithmetics of relational systems.Publ. Math. Debrecen 38 (1991), 39-48. MR 1100904 |
| . |
