Slaney, J.; Fujita, M.; Stickel, Mark E. Automated reasoning and exhaustive search: Quasigroup existence problems. (English) Zbl 0827.20083 Comput. Math. Appl. 29, No. 2, 115-132 (1995). The authors investigate the existence of some finite quasigroups using three different automated reasoning programs: DDPP, FINDER and MGTP. Their attention is concentrated on some \((i,j,k)\)-conjugate orthogonal idempotent Latin squares of order \(v\) (COILS(\(v\))) and quasigroups satisfying one of the identities: \(ab.ba=a\), \(ab.ba=b\), \((ba.b)b=a\), \(ab.b=a.ab\), \(a.ba=ba.b\), where the order is some \(v \geq 8\). The existence or nonexistence is proved by experimental evidence. The computations of such hard problems are very complicated. The programs DDPP, FINDER and MGTP are described, and in several cases the authors give their search times for comparison. Reviewer: E.Brožíková (Praha) Cited in 13 Documents MSC: 20N05 Loops, quasigroups 20-04 Software, source code, etc. for problems pertaining to group theory 05B15 Orthogonal arrays, Latin squares, Room squares 68T15 Theorem proving (deduction, resolution, etc.) (MSC2010) 68W30 Symbolic computation and algebraic computation Keywords:finite quasigroups; automated reasoning programs; orthogonal idempotent Latin squares; identities Software:FINDER PDFBibTeX XMLCite \textit{J. Slaney} et al., Comput. Math. Appl. 29, No. 2, 115--132 (1995; Zbl 0827.20083) Full Text: DOI References: [1] Fujita, M.; Slaney, J.; Bennett, F., Automatic generation of some results in finite algebra, (Proc. International Joint Conference on Artificial Intelligence (1993)), 52-57 [2] Bennett, F.; Zhu, L., Conjugate-orthogonal Latin squares and related structures, (Dinitz, J.; Stinson, D., Contemporary Design Theory: A Collection of Surveys (1992)) · Zbl 0765.05022 [3] Bennett, F., Quasigroup identities and Mendelsohn designs, Canadian J. Mathematics, 41, 341-368 (1989) · Zbl 0665.20035 [4] Baker, R., Quasigroups and tactical systems, Aequationes Mathematicae, 182, 96-303 (1978) · Zbl 0393.05017 [5] Mendelsohn, N., Combinatorial designs as models of universal algebras, Recent Progress in Combinatorics (1969), Academic Press: Academic Press New York · Zbl 0192.33302 [6] Zhang, J., Search for Idempotent Models of Quasigroup Identities, Typescript (1993), Institute of Software, Academia Sinica: Institute of Software, Academia Sinica Beijing [7] Pritchard, P., Algorithms for finding matrix models of propositional calculi, J. Automated Reasoning, 7, 475-487 (1991) · Zbl 0743.03003 [8] Bibel, W., Constraint satisfaction from a deductive viewpoint, Artificial Intelligence, 35, 401-413 (1988) · Zbl 0645.68112 [9] J. de Kleer, An improved incremental algorithm for generating prime implicates, In Proc. AAAI’92; J. de Kleer, An improved incremental algorithm for generating prime implicates, In Proc. AAAI’92 · Zbl 0944.68173 [10] Meglicki, G., Sickel’s Davis-Putnam Engineered Reversely (1993), Anonymous ftp, arp.anu.edu.au, Canberra [11] Slaney, J., FINDER, finite domain enumerator: Version 2.0 notes and guide, (Technical Report TR-ARP-1/92 (1992), Automated Reasoning Project, Australian National University) [12] B. McKay and S. Radziszowski, R(5,4) = 25, Journal of Graph Theory; B. McKay and S. Radziszowski, R(5,4) = 25, Journal of Graph Theory [13] B. McKay and S. Radziszowski, Linear programming in some Ramsey problems, J. Combinatorial Theory, Series B; B. McKay and S. Radziszowski, Linear programming in some Ramsey problems, J. Combinatorial Theory, Series B · Zbl 0811.05047 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.