A topology on latticeordered groups

We introduce the concept of the strongpositive cone in a latticeordered group \((G,\leq,\cdot)\) and define the continuous latticeordered group. We also investigate the \(C\)topology and bi\(C\)topology given on a latticeordered group. The main results obtained in this paper are as follows: (1) \((G,\leq,\cdot)\) is a continuous latticeordered group if and only if \((G,\leq)\) is a continuous poset; (2) for the bi\(C\)topology \(\tau\) in a continuous latticeordered group \((G,\leq,\cdot)\), \((G,\cdot,\tau)\) is a topological group and \((G,\leq,\tau)\) is a topological lattice.