For linearly orderable monoids, quasi-ACCP and almost ACCP ascend to finitary power monoids, while atomicity, near atomicity, and quasi-atomicity do not; atomic power monoids are characterized as those from atomic MCD-monoids.
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On finitary power monoids of linearly orderable monoids
For linearly orderable monoids, quasi-ACCP and almost ACCP ascend to finitary power monoids, while atomicity, near atomicity, and quasi-atomicity do not; atomic power monoids are characterized as those from atomic MCD-monoids.