By John M. Erdman

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We denote the supremum of a set A by either sup A or A. When A = {a1 , . . , an } is finite we usually write a1 ∨ a2 ∨ · · · ∨ an A. or nk=1 ak for A. If A = {a1 , a2 , . . } is denumerable, we may write ∞ k=1 ak for The “sup” in “sup A” is pronounced like the English word “soup” not like the English word “sup” (to dine). The plural of “infimum” is “infima”; the plural of “supremum” is “suprema”. A subset of a partially ordered set may have many upper bounds, but the definite article in the definition of “supremum” requires justification.

An element of a monoid can have at most one inverse. 17. Proposition. If a is an invertible element of a monoid, then a−1 is also invertible and a−1 −1 = a. 18. Proposition. If a and b are invertible elements of a monoid, then their product ab is also invertible and (ab)−1 = b−1 a−1 . 16 a bit. 19. Proposition. If an element c of a monoid has both a left inverse c l and a right inverse c r , then it is invertible and c−1 = c l = c r . Hint for proof . 1. 20. Corollary. Let a and b be elements of a monoid.

1. Definition. Let ≤ be a partial ordering on a set S and A ⊆ S. An element l ∈ S is a lower bound for A if l ≤ a for all a ∈ A. In this case we also say that A is bounded below (or minorized) by l. Similarly, an element u ∈ S is an upper bound for A if u ≥ a for all a ∈ A. In this case we also say that A is bounded above (or majorized or dominated) by u. The element l is the greatest lower bound (or infimum) of A if (i) l is a lower bound for A, and (ii) if m is any lower bound for A, then l ≥ m.