Full Idea
A 'proper or improper part' is expressed by 'x < y', read as 'x is (a) part of y'. The relatively minor deviation from normal usage (of including an improper part, i.e. the whole thing) is warranted by its algebraical convenience.
Gist of Idea
Proper or improper part: x < y, 'x is (a) part of y'
Source
Peter Simons (Parts [1987], 1.1.02)
Book Reference
Simons,Peter: 'Parts: a Study in Ontology' [OUP 1987], p.11
A Reaction
Including an improper part (i.e. the whole thing) is not, Simons points out, uncontroversial, because the part being 'equal' to the whole is read as being 'identical' to the whole, which Simons is unwilling to accept.