We present in this chapter, first, the theory of partial order. One formulation is based on a weak partial order a ≤ b and another one on a strict partial order a < b. The latter theory is problematic because of the absence of any easy definition of equality. Next, we present lattice theory and give a short, self-contained proof of the subterm property. By this property, we get a solution of the word problem for finitely generated lattices. It also follows that lattice theory is conservative over partial order for the problem of derivability of an atom from given atoms.
In Section 4.3, the most basic structure of algebra, namely a set with an equality and a binary operation, is treated. The proof of the subterm property for such groupoids is complicated by the existence of a unit of the operation. The treatment can be generalized to operations with any finite number of terms.
It is possible to modify the rules of lattice theory so that they contain eigenvariables. The number of rules drops down to four instead of six (plus the two of partial order). Moreover, the subterm property has an almost immediate proof. We consider also a formulation of strict order with eigenvariable rules, which permits the introduction (in a literal sense) of a relation of equality. A normal form for derivations and some of its consequences such as the conservativity of strict order with equality over the strict partial order fragment and the subterm property are shown.