Vol. 15, No. 1, 1965

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Topological semilattices on the two-cell

Dennison Robert Brown

Vol. 15 (1965), No. 1, 35–46

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 × 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,{y : x y} 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 {y : x y} connected for each element x if and only if it is the continuous homomorphic image of I × I. Also, if {y : x y} is connected for each element x, then S, a TSL on the two-cell, is generated by its boundary B in the sense that B2 = 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 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 A if and only if {y : x y} is connected. Then (1) x 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
Received: 12 March 1964
Published: 1 March 1965
Dennison Robert Brown