# How did the definition of each “ordered set” come about?

I could get a little intuition about preset, poset, and toset. e.g. A toset is, in effect, a linearly ordered set, and a poset is a set in which no more than one element in the same order exists, although not all elements need to be comparable.

But I thought, if I knew where this definition was initially needed and how it appeared, I could get a better understanding. And it was hard for me to find out enough explanation. I'd appreciate it if you could let me know about this. In addition, if you have any references to the history of other ordered sets, please recommend it.

• A reference may be: G. Birkhoff, Lattice Theory. The beginning there goes over "partially ordered set" (written "poset" by those short of space) and its history. It also discusses many example of partially ordered sets that are not totally ordered, illustrating how useful that concept is. – Gerald Edgar May 17 at 14:20
• @Gerald Edgar: Birkhoff's book might be a bit heavy for what the OP appears to be interested in, but +1 nonetheless for a well documented text (by someone I know you knew). I'm busy with some stuff now (contract work), but if I get a chance later, I'll see if I can think of something else. But for now I'll mention that the 3rd edition (last edition, from 1967; corrected reprint in 1979) omitted a lot of the bibliographic references from the earlier editions. Fortunately, the 1948 2nd edition is freely available. – Dave L Renfro May 17 at 16:56
• For the early history of lattice theory, which is closely linked to the development of ordered set notions, see On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others by Dirk Schlimm (2011; freely available pre-publication version). – Dave L Renfro May 24 at 13:00
• Regarding the rapid rise in the study of lattices in the late 1920s to late 1930s, see the following two freely available survey/overview papers (both are written versions of principal addresses to a conference on lattice theory in April 1938): Lattices and their applications by Garrett Birkhoff (1938) and The representation of Boolean algebras (1938) by Marshall Harvey Stone (1938). – Dave L Renfro May 26 at 6:24