Category theory as coherently constructive lattice theory : an illustration

Dijkstra and Scholten have formulat.ed a theorem st.at.ing that all disjunctivity properties of a predicat.e tral1sforlll('r are l>l'('served by t.he const.ruct.ion of least prefix points. An alternative proof of t.heir t.heorem is present.ed based on two fundamental fixed point theorems, t.he abst.ract.iol1 t.heorem and t.he fusion t.heorem, and the fact that suprema in a lattice arc ciefilH'd by a Galois connect.ion. The abstraction theorem seems to be new; t.he fusion t.heor('m is known but. it.s import.ance does not seem to be fully recognised. The abstract.ion t.heorem, t.he fusion t,h('orem, aud Dijkst.ra and Scholt.en's theorem are then generalised t.o the ('ont('xt of cat.egory t.heory and shown to be valid. None of the t.heorems in t.his cout.ext. seems t.o be kuowu, alt.hough specific inst.ances of Dijkstra and Seholt.en's t.heorem ,\l'P known. The main point. of the papPI' is t.o discuss the process of drawing inspiration from lattiee theory t.o formulat.e t,lworems ill cat.egory t.h('ory (first. advocated by Lambek in 1968). We advance the view t.hat., ill ord('r t.o ('ont.ribut.e to the development of programming met.hodology, cat.('gory t.lwory lllay \)(' profi t.ably regarded as "constructive" lattice theory in which t.1H' add('d valliI' li('s ill ('st.ablishing t.hat the constructions are "coherent.". This paper was specially prepared for I)l'Ps('nt.at.iou at. t.h(' meeting ofIFIP Working Group 2.3 (Programming Met.hodology), .luue H194. I-~uowledge of (elementary) lattice theory is assulIwd. Kuowkdgl' of ('at.('gory t.heory is 110t.

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