TY - JOUR

T1 - Topologies on products of partially ordered sets III

T2 - Order convergence and order topology

AU - Erné, Marcel

PY - 1981/12

Y1 - 1981/12

N2 - This paper deals with the question under which circumstances filter-theoretical order convergence in a product of posets may be computed componentwise, and the same problem is treated for convergence in the order topology (which may differ from order convergence). The main results are: (1) Order convergence in a product of posets is obtained componentwise if and only if the number of non-bounded posets occurring in this product is finite (1.5). (2) For any product of posets, the projections are open and continuous with respect to the order topologies (2.1). (3) A product L of chains L i has topological order convergence iff all but a finite number of the chains are bounded. In this case, the order topology on L agrees with the product topology (2.7). (4) If (L i :j ∈J) is a countable family of lattices with topological order convergence and first countable order topologies then order topology of the product lattice and product topology coincide (2.8). (5) Let P 1 be a poset with topological order convergence and locally compact order topology. Then for any poset P 2, the order topology of P 1⊗P 2 coincides with the product topology (2.10). (6) A lattice L which is a topological lattice in its order topology is join- and meet-continuous. The converse holds whenever the order topology of L⊗L is the product topology (2.15). Many examples are presented in order to illustrate how far the obtained results are as sharp as possible.

AB - This paper deals with the question under which circumstances filter-theoretical order convergence in a product of posets may be computed componentwise, and the same problem is treated for convergence in the order topology (which may differ from order convergence). The main results are: (1) Order convergence in a product of posets is obtained componentwise if and only if the number of non-bounded posets occurring in this product is finite (1.5). (2) For any product of posets, the projections are open and continuous with respect to the order topologies (2.1). (3) A product L of chains L i has topological order convergence iff all but a finite number of the chains are bounded. In this case, the order topology on L agrees with the product topology (2.7). (4) If (L i :j ∈J) is a countable family of lattices with topological order convergence and first countable order topologies then order topology of the product lattice and product topology coincide (2.8). (5) Let P 1 be a poset with topological order convergence and locally compact order topology. Then for any poset P 2, the order topology of P 1⊗P 2 coincides with the product topology (2.10). (6) A lattice L which is a topological lattice in its order topology is join- and meet-continuous. The converse holds whenever the order topology of L⊗L is the product topology (2.15). Many examples are presented in order to illustrate how far the obtained results are as sharp as possible.

UR - http://www.scopus.com/inward/record.url?scp=51249186144&partnerID=8YFLogxK

U2 - 10.1007/BF02483819

DO - 10.1007/BF02483819

M3 - Article

AN - SCOPUS:51249186144

VL - 13

SP - 1

EP - 23

JO - Algebra universalis

JF - Algebra universalis

SN - 0002-5240

IS - 1

ER -