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An ordered regular semigroup, , is said to be principally ordered if for every there exists . A principally ordered regular semigroup is pointed if for every element, we have . Here we investigate those principally ordered regular semigroups that are eventually pointed in the sense that for all there exists a positive integer, , such that . Necessary and sufficient conditions for an eventually pointed principally ordered regular semigroup to be naturally ordered and to be completely simple aredoi:10.24200/squjs.vol24iss2pp139-146 fatcat:rwxirnyu3nbjvppecdqk6lbjb4