Semi-Lattices and Taxonomic Systems

First, in order to clear up some misconceptions which his paper might leave in the mind of the unwary reader, we shall outline his development. His habitual example is biological taxonomy which is characterized by Morton Beckner3 as finite (and small) number of classes of classes [taxonomic categories] . . . arranged in serial order from 1 to n. . .. The members of each category, called taxa, are classes of organisms. . . . Every [taxon Tj of the jth taxonomic category] (j < n) is included in a Tj+1; Tn is included in no taxon; no taxa are included in a T1. Thus biological taxonomic structures are rooted trees of a simple type wherein each of the paths from the root, Tn, to a terminal node, a taxon in T1, has the same length, i.e. comprise the same number of nodes. The justification of Thomason's axioms is, presumably, that in philosophy it is necessary to consider hierarchical systems which are more complex. In his development, he first observes that in any taxonomy