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

• Results about total semilattices can be found here.

Linguistic Note

The term total semilattice was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to make the terminology more compact.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.