Communications of the
Korean Mathematical Society CKMS

A fence is an ordered set that the order forms a path with alternating orientation. Let \(\bf F=\fen\) be a fence and let \(OT(\bf F)\) be the semigroup of all order-preserving self-mappings of \(\bf F\). We prove that \(OT(\bf F)\) is coregular if and only if \(|F|\leq 2\). We characterize all coregular elements in \(OT(\bf F)\) when \(\bf F\) is finite. For any subfence \(\bf S\) of \(\bf F\), we show that the set \(COT_S(\bf F)\) of all order-preserving self-mappings in \(OT(\bf F)\) having \(S\) as their range forms a coregular subsemigroup of \(OT(\bf F)\). Under some conditions, we show that a union of \(COT_S(\bf F)\)'s forms a coregular subsemigroup of \(OT(\bf F)\).

Keywords: order-preserving, fence, self-mapping, semigroup, coregular

MSC numbers: 0M20, 20M05, 20M17