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Let A denote the class of all aperiodic monoids with central idempotents. A description of all Cross subvarieties of A, based on identities that they satisfy and monoids that they cannot contain, is given. The two limit subvarieties of A, published by Marcel Jackson in 2005, turn out to be the only finitely generated, almost Cross subvarieties of A. It follows that it is decidable in quartic time if a finite monoid in A generates a Cross variety.
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Edmond W. H. Lee, Cross varieties of aperiodic monoids with central idempotents. Port. Math. 68 (2011), no. 4, pp. 425–429