On Partial Orders in Complex Fuzzy Logic

Abstract

A partial ordering $\preceq$ on the unit circle of the complex plane ${\bf D}$ induced by the algebraic product was introduced and it was proved that $({\bf D},\preceq)$ is a complete lattice (Dick, 2005). In this letter, it is pointed out that the relation $\preceq$ is not a partial order on ${\bf D}$ and that $({\bf D},\preceq)$ is not a lattice. It is shown that the relation $\preceq$ is a partial order over ${\bf D}^{\circ }\cup \lbrace 1\rbrace$ and a strict partial order over ${\bf D}^{\circ }-\lbrace 0\rbrace$ , where ${\bf D}^{\circ }$ is the interior of ${\bf D}$ .