Topological semilattices on the two-cell

Dennison Robert Brown

Vol. 15 (1965), No. 1, 35–46

DOI: 10.2140/pjm.1965.15.35

Abstract

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 two-cell 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 two-cell 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 two-cell, where the choice of domain depends on the location of the zero element.

It is also proved that a TSL on the two-cell 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 two-cell, 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 sub-TSL of $S$.

Mathematical Subject Classification

Primary: 54.56

Milestones

Received: 12 March 1964

Published: 1 March 1965