Article
Full entry |
PDF
(0.7 MB)
Feedback
PDF
(0.7 MB)
Feedback
Keywords:
cardinal sum; cardinal product; ordinal sum; ordinal product; $n$-ary relational system; $n$-ary ordered set; cardinal power; ordinal power
cardinal sum; cardinal product; ordinal sum; ordinal product; $n$-ary relational system; $n$-ary ordered set; cardinal power; ordinal power
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.
References:
[1] G. Birkhoff: An extended arithmetic. Duke Math. J. 3 (1937), 311-316. DOI 10.1215/S0012-7094-37-00323-5 | MR 1545989 | Zbl 0016.38702
[2] G. Birkhoff: Generalized arithmetic. Duke Math. J. V (1942), 283-302. DOI 10.1215/S0012-7094-42-00921-9 | MR 0007031 | Zbl 0060.12609
[3] G. Birkhoff: Lattice Theory. Amer. Math. Soc., Providence, Rhode Island, Third Edition, 1973. MR 0227053
[4] M. M. Day: Arithmetic of ordered systems. Trans. Amer. Math. Soc. 58 (1945), 1-43. DOI 10.1090/S0002-9947-1945-0012262-4 | MR 0012262 | Zbl 0060.05813
[5] V. Novák: On a power of relational structures. Czechoslovak Math. J. 35 (1985), 167-172. MR 0779345
[6] V. Novák M. Novotný: Binary and ternary relations. Math. Bohem. 117(1992), 283-292. MR 1184541
[7] V. Novák M. Novotný: Pseudodimension of relational structures. Czechoslovak Math. J. (submitted). MR 1708362
[8] J. Šlapal: Direct arithmetics of relational systems. Publ. Math. Debrecen 38 (1991), 39-48. MR 1100904
Search
Partner of
