Context: self-education.
I am currently getting my head round semilattices.
My understanding is that a semilattice $(S, \odot)$ is a semigroup whose operation $\odot$ is both commutative and idempotent, that is:
- $\forall x, y \in S: x \odot y = y \odot x$
- $\forall x \in S: x \odot x = x$
One of the first things you find out is that $\odot$ induces an ordering $\preccurlyeq$ defined as: $$x \preccurlyeq y \iff x \odot y = y$$
from which it follows after some straightforward algebra that:
$$x \odot y = \sup_\preccurlyeq \{x, y\}$$
and lo and behold, we have a join semilattice.
ASIDE: There's very much a chicken-and-egg thing going on here. That is, from what I can tell from the literature, often the join semilattice is defined directly on an ordered set, from which the "join" operation is then defined as $x \vee y := \sup \{x, y\}$, rather than starting with the abstract semilattice and then creating the operation on that.
But there is an added level of interest when $(S, \odot)$ is such that every subset of $S$ is a subsemilattice of $S$, that is, every subset is closed under $\odot$. Because in that case, the ordering induced on $(S, \odot)$ is total. And conversely, the join operation defined on a set by a total ordering is such that every subset is closed under that join operation.
ASIDE: I note that the usual symbol for the operation here is $\vee$, for "join", or $\wedge$ for "meet", and a distinction is usually made between the two, where the ordering induced by $\wedge$ is the other way round from the one induced by $\vee$, but as these things are dual, the distinction is arguably less important at the most abstract level than might otherwise be inferred. Hence my "neutral" notation $\odot$.
My question is:
Is there a special name for such a semilattice - one whose subsets are all subsemilattices? Or do we have to resort to cumbersome verbal constructs like "Let $(S, \odot)$ be a semilattice whose subsets are all closed under $\odot$" or "Let $(S, \odot)$ be a semilattice whose subsets are all subsemilattices"?
EDIT: I wonder whether the world would object if I were to suggest the neologism total semilattice. This would neatly dovetail with the fact that the ordering it induces is a total ordering.
In terms of ordered sets, it is called a linear order. In terms of semigroups, I would call it a chain of idempotents.