Topological lattices on the
$n$cell
have been studied by L. W. Anderson, A. D. Wallace, A. L. Shields, and L. E. Ward,
Jr. In particular, these authors have papers setting forth conditions under which a
topological lattice on the twocell is topologically isomorphic to the product lattice
$I\times I$.
The primary purpose of this paper is the investigation of topological
semilattices (commutative, idempotent topological semigroups) on the
twocell which retain the lattice like property that for each element
$x,\left\{y:x\leqq y\right\}$ is a
connected set. Specifically, it is shown that any such entity is the continuous
homomorphic image of one of a fixed pair of semilattices on the twocell, where the
choice of domain depends on the location of the zero element.
It is also proved that a TSL on the twocell has an identity (a unique maximal element) and
$\left\{y:x\leqq y\right\}$ connected for
each element
$x$
if and only if it is the continuous homomorphic image of
$I\times I$. Also, if
$\left\{y:x\leqq y\right\}$ is connected for
each element
$x$,
then
$S$,
a TSL on the twocell, is generated by its boundary
$B$ in the
sense that
${B}^{2}=S$.
Semilattices on the
$n$cell are
also discussed. Let
$S$ be such
an object with boundary
$B$.
It is proved that if
$x$ is a
maximal element of
$S$,
then
$x\in B$. If
$S$ has an identity, 1,
and
$T$ is a continuum
chain from 1 to
$0$,
then
$S=BT$.
Finally, let
$S$ be a continuum
TSL with 1 and let
$A$ be
the subset defined by
$x\in A$
if and only if
$\left\{y:x\leqq y\right\}$ is
connected. Then (1)
$x\in A$
if and only if there is a continuum chain from 1 to
$x$; and (2)
$A$ is a nondegenerate
continuum subTSL of
$S$.
