Quadrics via Semigroups

Let M2 be the semigroup of linear endomorphisms of a plane. We show that the space of idempotents in M2 is a hyperboloid of one sheet, the set of semigroup-theoretic inverses of a nonzero singular element in M2 is a hyperbolic paraboloid, and the set of nilpotent elements in M2 is a right circular cone. This is the story of the rediscovery of classical three-dimensional geometry, especially the geometry of quadric surfaces, while studying the semigroup M2(R) of linear endomorphisms of a real plane. One of the surfaces that appears prominently in this context is the hyperboloid of one sheet, referred to as spaghetti bundle in [8]. In this story the spaghetti presents itself as the set of idempotents in M2(R), the cone emerges as the set of nilpotent elements and the hyperbolic paraboloid as the set of semigroup-theoretic inverses of a singular element. This rediscovery was briefly announced in [5]. Generalizations of some of the ideas presented here to semigroups of linear endomorphisms of higher dimensional vector spaces are discussed in [6]. The little bit of semigroup theory quoted below is based on [3]. 1. The Semigroup M2(R) A set S together with an associative binary operation in S is called a semigroup. An element e in S is called an idempotent if e = e. IfX ⊆ S, the set of idempotents in X is denoted by E(X). In any semigroup S we can define certain equivalence relations, denoted by L , R , J , D and H , and called Green’s relations. Let S denote S if S has an identity element. Otherwise, let it denote S with an identity element 1 adjoined. For a, b ∈ S the first three are defined by aL b ⇔ aS = bS aR b ⇔ Sa = Sb aJ b ⇔ SaS = SbS and the remaining ones by D = L ◦ R = R ◦ L , H = L ∩ R . If a ∈ S, the set of all elements in S which are L -equivalent to a is denoted by La and is called the L -class containing a. The notations Ra, Ja, Da, Ha have similar meanings. These are the Green classes in S. Two elements a, a′ in a semigroup S are called inverse elements if aa′a = a, a′aa′ = a′. A semigroup in which every element has an inverse is called a regular semigroup. A well-known example of a regular semigroup is Mn(K) (where K = R, or K = C) of linear endomorphisms of an n-dimensional vector space V over K under 1991 Mathematics Subject Classification. 15A04, 20M17, 51N25.