Algebre: Exercises avec solutions by E Ramis

By E Ramis

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The following establishes a similar connection between the congruences on R and ideals of R: (a) If θ € Con R then 0④θ is an ideal of R, and for a, b € R we have ①a, b② € θ iff a ✁ b € 0④θ; (b) If I is an ideal of R then the binary relation θ defined on R by ①a, b② € θ iff a ✁ b € I is a congruence on R with 0④θ ✏ I. Thus the mapping θ ÞÑ 0④θ is an order-preserving bijection between congruences on R and ideals of R. These two examples are a bit misleading in that they suggest any congruence on an algebra might be determined by a single equivalence class of the congruence.

1) The lattice of subsets of a set is an algebraic lattice (where the compact elements are finite sets). (2) The lattice of subgroups of a group is an algebraic lattice (in which “compact” = “finitely generated”). (3) Finite lattices are algebraic lattices. (4) The subset r0, 1s of the real line is a complete lattice, but it is not algebraic. In the next chapter we will encounter two situations where algebraic lattices arise, namely as lattices of subuniverses of algebras and as lattices of congruences on algebras.

T. Schmidt [1963] B. J´onsson [1953] P. Pudl´ak [1976] E XERCISES §5 1. 2). 2. 2). 42 II The Elements of Universal Algebra 3. Show that the normal subgroups of a group form an algebraic lattice which is modular. 4. Show that every group and ring is congruence-permutable, but not necessarily congruencedistributive. 5. Show that every lattice is congruence-distributive, but not necessarily congruencepermutable. 6. 5, verify that subuniverses of the new algebra are precisely the congruences on A. 7.

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