Exercise 13, § 1

a) A monoid M is said to be lower semi-lattice ordered (or simply semi-lattice ordered) if it is an ordered monoid, if inf(x, y) exists for all x, y ∈ M, and if

for all x, y, z ∈ M. Prove the identities:

From the first of these relations, deduce that $x \leq z$ and $y \leq z$ imply $x + y \leq z + \inf(x, y)$.

Show that Prop. 11 and Cor. (VI, pp. 14-15) are valid in a semi-lattice ordered monoid. b) Let $M$ be a semi-lattice ordered monoid and $N$ a submonoid such that $\inf_M(x, y) = \inf_N(x, y)$ for $x, y \in N$ and such that the elements of $N$ are invertible in $M$. Show that the subgroup $G$ of $M$ consisting of elements $x - y$ for $x, y \in N$, is lattice ordered.