Adjoining Identities and Zeros to Semigroups

This note shows how iteration of the standard process of adjoining identities and zeros to semigroups gives rise naturally to the lexicographical ordering on the additive semigroups of n-tuples of nonnegative integers and n-tuples of integers. 1. Semigroups with identities and zeros A binary operation ∗ on a set S is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ S. A semigroup is a nonempty set with an associative binary operation ∗. The semigroup is abelian if a ∗ b = b ∗ a for all a, b ∈ S. The trivial semigroup S0 consists of a single element s0 such that s0 ∗ s0 = s0. Theorems about abstract semigroups are, in a sense, theorems about the pure process of multiplication. An element u in a semigroup S is an identity if u ∗ a = a ∗ u = a for all a ∈ S. If u and u are identities in a semigroup, then u = u ∗ u = u and so a semigroup contains at most one identity. A semigroup with an identity is called a monoid. If S is a semigroup that is not a monoid, that is, if S does not contain an identity element, there is a simple process to adjoin an identity to S. Let u be an element not in S and let I(S, u) = S ∪ {u}. We extend the binary operation ∗ from S to I(S, u) by defining u ∗ a = a ∗ u = a for all a ∈ S, and u ∗ u = u. Then I(S, u) is a monoid with identity u. An element v in a semigroup S is a zero if v ∗ a = a ∗ v = v for all a ∈ S. If v and v are zeros in a semigroup, then v = v ∗ v = v and so a semigroup contains at most one zero. If S is a semigroup that does not contain a zero element, there is also a simple process to adjoin a zero to S. Let v be an element not in S and let Z(S, v) = S ∪ {v}. We extend the binary operation ∗ from S to Z(S, v) by defining v ∗ a = a ∗ v = v for all a ∈ S, and v ∗ v = v. Then Z(S, v) is a semigroup with zero v. It is important to note that the process of adjoining an identity to a semigroup S is well-defined even if S contains an identity. Similarly, the process of adjoining a zero to a semigroup S is well-defined even if S contains a zero. The element s0 in the trivial semigroup S0 is both an identity and a zero. In this note we investigate what happens when we start with the trivial semigroup and add new identities and new zeros. Date: February 2, 2008. 2000 Mathematics Subject Classification. Primary 20M05, 20M14,11B75,11B99.