On the Multidimensionality of Alienation

Appendix II: A Theorem Theorem: A congruence class, C, of a semilattice is closed, convex, and contains a maximal element. (1) C is closed. Assume aEC and beC. Then abeCC. Now,a=aacCC, so C=CC. (2) C contains a maximal element. By closure the product, p, of all elements in C is in C, and by idempotence pc=p for an arbitrary ceC. Hence p is a maximal element of C. (3) C is convex. Let aeC, beC, a>x>b, xeX. xa-a so XC=C, xb-x so XC-X. Thus X =C.