A perspective on algebraic representations of lattices
This is a comprehensive survey of known results on algebraic representations of lattices. After a short historical overview of classical results by Birkhoff, Frink and Whitman, it continues with various attempts at the representation problem. At first, it presents the classical Birkhoff-Frink theorem giving the equivalence of algebraicity of a lattice \(L\) and its representability as \(\text{Sub } A\) for an algebra \(A\) with countable similarity type, which is equivalent to \(L\cong \text{Sub } A\) for a groupoid \(A\) or a commutative Moufang loop. It states the problem of characterizing \(\text{Sub } A\) for a semigroup \(A\). Next, he discuss known results on the representation of \(L\) as a sublattice of \(\text{Eq} (A)\). He quotes the theorems by B. Jónsson: Every \(L\) is isomorphic to a type-3 sublattice of some \(\text{Eq} (A)\). \(L\) has a type-2 partition-lattice representation iff \(L\) is modular. The case of type-1 representation and the property ``to be Arguesian'' is discussed in full detail. The following part is devoted to the famous Grätzer-Schmidt theorem: \(L\) is algebraic iff \(L\cong \text{Con } A\) for some algebra \(A\) (which can be of type 3). Pudlák's proof is considered and his construction is compared with that of the original one by Grätzer and Schmidt. Grätzer and the author proved that if \(L\) is a modular and algebraic lattice, then \(L\cong \text{Con } A\) with \(\text{Con } A\) a type-2 sublattice of \(\text{Eq} (A)\). It is a problem if this can be extended also for a type-1 representation. The next part contains results on the representation of a finite lattice \(L\) as \(\text{Con } A\) for a finite algebra \(A\). The paper lists results by Pálfy, Pudlák, Tůma and McKenzie. The author lists also known results on representations \(L\cong \text{Con } A\) for \(A\) of finite (or countable) similarity type. For uncountable similarity type, he mentions \(k\)-algebraicity of \(L\) for some cardinal \(k\). Although the theorem by Tischendorf and Tůma characterizing any distributive algebraic lattice as \(\text{Con } L\) for some lattice \(L\) is known since summer 1993, it is only mentioned here as a note added in proof. Hence, the chapter on congruence lattices of lattices is too brief to give a lucid overview of this topic. It also does not contain any of E. T. Schmidt's results on this topic. The paper contains also a brief outline of the lattices of equational theories. The last part is devoted to a very detailed discussion of Pudlák's construction of yeast graphs and connected concepts.
scientific article; zbMATH DE number 590885
scientific article