On the atoms of algebraic lattices arising in 𝔮-theory

Abstract

We determine many of the atoms of the algebraic lattices arising in q-theory of finite semigroups.Caution 3.4. In an algebraic lattice L, AtomspLq can be empty.Corollary 3.5. If L has no atoms, then rB, ℓ 2 s has no atoms.Remark 3.6. In an algebraic lattice L, the atoms of rℓ 1 , Ts are the covers of ℓ 1 in L, so in general, they are unrelated to AtomspLq.Principle 3.7 (No atoms for L, an algebraic lattice). If each compact element c ‰ B has a compact element other than B strictly below, then AtomspLq " ∅, and conversely, since the atoms are the compact covers.Principle 3.7 was used in [9, Proposition 7.1.24] to prove the following proposition.Proposition 3.8. The algebraic lattice CntpPVq has no atoms.As a consequence, we can prove that CC has no atoms.Proposition 3.9. The algebraic lattice CC has no atoms.Proof. By [9, Theorem 2.3.9], there is a surjective map q : CC Ñ CntpPVq preserving all sups and finite meets. The bottom of CntpPVq is the constant map to the trivial pseudovariety. In [9, Page 121] it is shown that each constant map has a unique preimage under q, hence if R is not the bottom of CC, then is does not map to the bottom of CntpPVq under q. Since CntpPVq has no atoms, we can find B ‰ α ă Rq. By surjectivity, there exists S with Sq " α. Since q preserves finite infs, we obtain pR X Sqq " α and so R X S ă R and R X S is not the bottom. Thus CC has no atoms.The reader is referred to [9, Proposition 2.1.11] for the definition of CC and [9, Page 75] for the definition of PVRM. Fact 3.10. If D denotes the class of all divisions, then( Proof. One way of calculating CC isCCpXq " tf | f Ď s d 1 pf 1ˆ¨¨¨ˆfn qd 2 , d 1 , d 2 P D, f i P Xu (see Proposition 2.1.14 in [9]) from which (1) follows. Also CCp1 V | V P PVq is closed under Axiom (co-re), as D is, so is a pseudovariety of relational morphisms in PVRM (see Proposition 2.1.8(c) in [9]) proving (2).Definition 3.11.CntpPVq´" tα P CntpPVq | α ď 1 PV u CC´" tβ P CC | β ď Du Fact 3.12. pCC´qq " CntpPVqṔ roof. Since Dq " 1 PV and q is order preserving pCC´qq Ď CntpPVq´. If α P CntpPVq´, then since α P CntpPVq, there exists R P CC, so Rq " α. Thus since q preserves finite intersections (intersection equals meet) pR X Dqq " α and R X D P CC´.Corollary 3.13. The lattices CC, CC´, CntpPVq, CntpPVq´have no atoms.