Estimating evolutionary relationships is basic to the objectives of systematics. Comparative data, structured as taxonomic characters, are usually the esssential considerations on which such estimates are founded. Some taxonomic characters are more useful than others for structuring plausible estimates of evolutionary relationship. Thus, one of the primary challenges to the systematist is the construction of taxonomic characters most useful for this purpose. Since taxonomic characters are the result of action on the part of the systematist, they must be defined operationally. It is our hope, however, that these operationally defined characters will conform to an ideal that, itself, cannot be operationally defined insofar as the concept depends on history that is, usually, inherently unknowable. It is, nonetheless, essential to our conceptual methods that this ideal concept be well defined. Here we present a series of definitions leading to a clear ideal concept of irue cladistic character. This series includes definitions of the concepts: evolutionary unit, qualitative taxonomic character, monophyletic group, divergent character, true cladistic character, operational cladistic character, and the post-factum ideal relation between an operational cladistic character and an estimate of cladistic history. A concise characterization of true cladistic character is presented and proved.

Using formal algebraic definitions of "cladistic character" and "character compatibility", the concept of "binary factors of a cladistic character" is formalized and used to describe and justify an algorithm for checking the compatibility of a set of characters. The algorithm lends itself to the selection of maximal compatible subsets when compatibility fails.

Introduction that these definitions and result:; art' an accurate reflection of current practice in this field. For further biologic.:~! b&ckground and motivation we refer the reader to rcferenccs [ l-4]. 2. Definitions and results WC supp~~se throughout that all sets are finite. El 1's are evolutionary units. Definition 2.1. A free posts is a partially ordered set having the prrlperty that Q s c I and b 5 c together imply c1 s b or II G tz. A trw senddce is c"s tree posct in which an! two elcmtSntc TV :md b have ;I greatest lower bound. denoted 17 4 b. in what follows. S will denote a fixed set of EW's under study and S' WIII rq~re\cnt an estirmtc of S', the (unknown) true evolutionary history of S. By taking s 7 ! to mean b*.x is an axxstor of y " wt' view S as a tree posct, S' and S * as tree s;s;nilattices in which s A. y is the most rtxxnt common ancestor of s and y. Definition 2 .2, A clatiik-chumctcr on S is a map K : S-P ;ind a cidistic c'hrlrcACtdf OfI s * is an onto map K : S*-+ P where P is a tree semilattice (the Cl~clrclCWr ottrtc twc 1. Definition 2.3. Let S* hr ;1 tree scmilattil'c. A cladistic character h' : S *-P is frrc~ If and only if K satisfies the following three conditions for 11. b E S *: (i) hi E K '(K(G)) where n" = .q K '(K(ri)) (ii) L! 5 h implies K(d) s K(h) (iii) K(Lz) 5.~ K(b) implic~ 5 s b. Definitions 2 and 3 are discussed in some detail in [.J] and WC will simply

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In this paper a coordinating semigroup is used to define and characterize certain homomorphisms on a bounded poset or semilattice. These homomorphisms are determined by their kernels and in the semilattice case the ideals which occur as such kernels are characterized.1. Introduction. In [4] B. J Thorne characterized certain congruence relations on a bounded lattice by looking at AP homomorphisms on a coordinatizing Baer semigroup. We intend to carry out a similar procedure for bounded posets and semilattices. It will turn out that one of our semilattice results gives Thome's central result as a corollary. Our notation will be that of [4] If £ is a semigroup with 0 and ASS we define L(A) = {x e S; xa = 0 for all a e A}, R{A) = {χeS;ax = 0 for all a e A], LR{A) = L(R(A)), RL(A) = R(L(A)), and so forth. If xeS we write L({x}) = L(x) and R({x}) = R(x). We define £f(S) = {L(x);xeS} and &(S) = {R(x);xeS} and say that S coordinatizes a poset P in case P = ^f(S) when Sfiβ) is partially ordered by set inclusion.The coordinatization machinery which we will use is developed in [2]. The following is a summary of the relevant material.

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