Lattice isomorphisms of commutative semigroups have been investigated from different points of view, and results relating to this topic are distributed into distinct chapters. We recall that, in Chapter X, commutative non-periodic cancellative semigroups are considered (and exhaustive information on lattice isomorphisms of these semigroups has been obtained). Commutative semigroups are one of the objects of attention in Chapter XIII; they are also touched upon in the exercises for Chapter XIV. The present chapter, as can be seen from the title, is entirely devoted to them. Here a starting motif is the semilattice decompostion of an arbitrary commutative semigroup into archimedean components. It is clear that examination of lattice isomorphisms of arbitrary commutative semigroups has to include the case of archimedean semigroups. Section 35 is (almost all) devoted to this case. A key result of this section states that the class of commutative archimedean semigroups is lattice-closed in the class of all commutative semigroups. In particular, one can deduce from this fact that, under a lattice isomorphism of a commutative semigroup S upon a commutative semigroup T, the archimedean components of S are mapped onto those of T, i.e., according to the terminology of Subsection 31.12, any commutative semigroup is semilattice-projectable within the class of all commutative semigroups. By Observation 31.12, the last fact, in its turn, shows that another case of principle—arising when investigating lattice isomorphisms of arbitrary commutative semigroups—is the case of semilattices. Section 36 is just devoted to the projectivities of semilattices.
KeywordsCommutative Semigroup Lattice Isomorphism Archimedean Semigroup Proper Divisor Ordinal Component
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