Some Observations on Oligarchies, Internal Direct Sums, and Lattice Congruences
A set-theoretic abstraction of some deep ideas from lattice theory is presented and discussed. By making use of this abstraction, many results from seemingly disparate disciplines can be examined, proved, and subtle relationships can be discovered among them. Typical applications might involve decision theory when presented with evidence from sources that yield conflicting optimal advice, insights into the internal structure of a finite lattice, and the nature of homomorphic images of a finite lattice. Some needed historical background is provided. (Presented in conjunction with the volume dedicated to the 70th Birthday celebration of Professor Boris Mirkin.) In particular, there is a connection to some early work of Mirkin (On the problem of reconciling partitions. In: Quantitative Sociology, International Perspectives on Mathematical and Statistical Modelling, pp. 441–449. Academic, New York, 1975).
KeywordsOligarchy Lattice congruence Simple lattice Residual mapping
The author wishes to thank Professors Bruno Leclerc, Bernard Monjardet, and Sandor Radeleczki for commenting on earlier versions of the manuscript. Their remarks were a big help. Section 2 especially was revamped because of suggestions from Professor Radeleczki. Thanks are also given to an anonymous referee for many suggestions involving style and clarity of exposition.
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