Definition:Total Semilattice

Definition
Let $\struct {S, \odot}$ be a semilattice.

Let $\struct {S, \odot}$ have the property that every subset of $\struct {S, \odot}$ is a subsemilattice.

That is, such that every subset of $\struct {S, \odot}$ is closed under $\odot$.

Then $\struct {S, \odot}$ is known as a total semilattice.

Also see

 * Total Semilattice has Unique Total Ordering, justifying the name