In 1914, Norbert Weiner introduced the concept of a semiorder. In 1956, Duncan Luce brought semiorders to economics to study preferences when indifference is not transitive. In the next 70 years, semiorders have become a key concept for modeling economic choices and decisions when risks or errors are present. In 1958, Dana Scott and Patrick Suppes gave sufficient conditions for a semiorder to be described by means of a threshold of perception. In the decades that followed, others gave various necessary conditions. In 2016, A. Giarlotta and I gave a description of all semiorders using a kind of shifted lexicographic ordering that extends the notion of a threshold of perception. In 2025, A. Giarlotta, D. Carpentiere, and I have shown that this description enables much shorter and intuitive proofs of most of the principal theoretical results about semiorders.

In this talk, we will explain semiorders, review their history, and sketch the ideas we developed in the description of all semiorders.