Previous |  Up |  Next


groupoid; unification
Given a groupoid $\langle G, \star \rangle $, and $k \geq 3$, we say that $G$ is antiassociative if an only if for all $x_1, x_2, x_3 \in G$, $(x_1 \star x_2) \star x_3$ and $x_1 \star (x_2 \star x_3)$ are never equal. Generalizing this, $\langle G, \star \rangle $ is $k$-antiassociative if and only if for all $x_1, x_2, \ldots , x_k \in G$, any two distinct expressions made by putting parentheses in $x_1 \star x_2 \star x_3 \star \cdots \star x_k$ are never equal. \endgraf We prove that for every $k \geq 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.
[1] Baader, F., Snyder, W.: Unification theory. Handbook of Automated Reasoning A. Robinson et al. North-Holland/Elsevier, Amsterdam, MIT Press Cambridge 445-533 (2001). DOI 10.1016/B978-044450813-3/50010-2 | Zbl 1011.68126
[2] Braitt, M. S., Hobby, D., Silberger, D.: Completely dissociative groupoids. Math. Bohem. 137 (2012), 79-97. MR 2978447 | Zbl 1249.20075
[3] Braitt, M. S., Silberger, D.: Subassociative groupoids. Quasigroups Relat. Syst. 14 (2006), 11-26. MR 2268823 | Zbl 1123.20059
[4] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra. Graduate Texts in Mathematics 78 Springer, New York (1981). DOI 10.1007/978-1-4613-8130-3_3 | MR 0648287 | Zbl 0478.08001
[5] Drápal, A., Kepka, T.: Sets of associative triples. Eur. J. Comb. 6 (1985), 227-231. DOI 10.1016/S0195-6698(85)80032-9 | MR 0818596 | Zbl 0612.05003
[6] Herbrand, J.: Recherches sur la théorie de la démonstration. Travaux de la Société des Sciences et des Lettres de Varsovie 33 128 pages (1930), French. MR 3532972 | Zbl 56.0824.02
[7] Huet, G. P.: Résolution d'équations dans des langages d'ordre $1,2,\dots,\omega$. Thèse d'État, Université de Paris VII (1976), French.
[8] Ježek, J., Kepka, T.: Medial groupoids. Rozpr. Cesk. Akad. Ved, Rada Mat. Prir. Ved 93 (1983), 93 pages. MR 0734873 | Zbl 0527.20044
[9] Knuth, D. E.: The Art of Computer Programming. Vol. 1: Fundamental Algorithms. Addison-Wesley Series in Computer Science and Information Processing Addison-Wesley, London (1968). MR 0286317 | Zbl 0191.17903
[10] Robinson, J. A.: A machine-oriented logic based on the resolution principle. J. Assoc. Comput. Mach. 12 (1965), 23-41. DOI 10.1145/321250.321253 | MR 0170494 | Zbl 0139.12303
[11] Stanley, R. P.: Enumerative Combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics 62 Cambridge University Press, Cambridge (1999). DOI 10.1017/CBO9780511609589 | MR 1676282 | Zbl 0928.05001
Partner of
EuDML logo