Subalgebra lattice

of a universal algebra $ A $

The partially ordered set $ \mathop{\rm Sub} A $( ordered by inclusion) of all subalgebras of $ A $. For any $ X, Y \in \mathop{\rm Sub} A $, their supremum is the subalgebra generated by $ X $ and $ Y $, and their infimum is the intersection $ X \cap Y $. An intersection of subalgebras may be empty, and therefore for certain types of algebras (e.g. semi-groups and lattices) one includes among the subalgebras the empty set. For any algebra $ A $, the subalgebra lattice $ \mathop{\rm Sub} A $ is algebraic, and, conversely, for any algebraic lattice $ L $ there is an algebra $ A $ such that $ L \cong \mathop{\rm Sub} A $( the Birkhoff–Frink theorem). Any lattice can be imbedded in the lattice $ \mathop{\rm Sub} A $ for some group $ A $.

The subalgebra lattice $ \mathop{\rm Sub} A $ is one of the basic derived structures associated with an algebra $ A $( together with such structures as the group of automorphisms, the semi-group of endomorphisms, the lattice of congruences, etc.). The problems involved in the study of the connections between algebras and their subalgebra lattices are divided into the following aspects: lattice isomorphisms, lattice characteristics of various classes of algebras, and research on algebras with various restrictions on the subalgebra lattice. Two algebras $ A $ and $ B $ are called lattice isomorphic if $ \mathop{\rm Sub} A \cong \mathop{\rm Sub} B $; an isomorphism between $ \mathop{\rm Sub} A $ and $ \mathop{\rm Sub} B $ is called a lattice isomorphism (or a projectivity) of $ A $ onto $ B $. Isomorphic algebras are lattice isomorphic but the converse is far from being necessarily true. An algebra $ A $ is said to be lattice definable (in a given class $ {\mathcal K} $) if for any algebra $ B $( from $ {\mathcal K} $) of the same type, $ \mathop{\rm Sub} B \cong \mathop{\rm Sub} A $ implies that $ B \cong A $. In some cases (e.g. in semi-groups) the concept of lattice definability is extended by adding to the conclusion of the implication the condition "or B is anti-isomorphic to A" , since anti-isomorphic semi-groups are also lattice isomorphic. The classic example of lattice definability is given by the first basic theorem of projective geometry (see [1]), where vector spaces over skew-fields are considered as $ A $. The following are also lattice definable: every Abelian group which contains two independent elements of infinite order, every free group (free semi-group) and every group (semi-group) which is non-trivially decomposable into a free product, every nilpotent group without torsion, every commutative semi-group with a cancellation law and without idempotents, every free semi-group of idempotents, and a free semi-lattice with more than two free generators. In this case it often turns out that each projectivity of an algebra is induced by an isomorphism of it (or by an anti-isomorphism). A class $ {\mathcal K} $ of algebras of the same type may contain algebras which are not lattice definable, but still have the property that $ A \in {\mathcal K} $ and $ \mathop{\rm Sub} B \cong \mathop{\rm Sub} A $ imply that $ B \in {\mathcal K} $; in this case $ {\mathcal K} $ is said to be lattice definable (or lattice closed); if here the algebras $ B $ are taken only from a class $ {\mathcal M} $, then "in the class M" is added to the corresponding term. Lattice-closed classes include the class of all solvable groups.

Many of the restrictions imposed on algebras are formulated in terms of subalgebra lattices. The classic examples are the minimum and maximum conditions for subalgebras. $ \mathop{\rm Sub} A $ satisfies the maximum condition if and only if all subalgebras in $ A $ are finitely generated (see also Group with a finiteness condition; Semi-group with a finiteness condition). Other restrictions imposed on subalgebra lattices include such lattice-theoretical properties as distributivity, modularity, various forms of semi-modularity, the Jordan–Dedekind condition, complementedness, relative complementedness, etc. For example, for a group $ A $ the subalgebra lattice is distributive if and only if $ A $ is locally cyclic (Ore's theorem); distributivity conditions are also studied in semi-groups, associative rings, modules, Lie algebras, etc.

In addition to isomorphisms of subalgebra lattices, dualities (i.e. anti-isomorphisms) and homomorphisms are also studied. When $ A $ is a topological algebra it is most natural to consider the lattice of all closed subalgebras of $ A $; active research is carried out on the corresponding problems.

References

[1]	R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952)
[2]	M. Suzuki, "Structure of a group and the structure of its lattice of subgroups" , Springer (1967)
[3]	L.A. Skornyakov, "Complemented modular lattices and regular rings" , Oliver & Boyd (1964) (Translated from Russian)
[4]	P.M. Cohn, "Universal algebra" , Reidel (1981)
[5]	L.E. Sadovskii, "Some lattice-theoretical problems in the theory of groups" Russian Math. Surveys , 23 : 3 (1968) pp. 125–156 Uspekhi Mat. Nauk , 23 : 3 (1968) pp. 123–157
[6]	M.N. Arshinov, L.E. Sadovskii, "Some lattice-theoretic properties of groups and semigroups" Russian Math. Surveys , 27 : 6 (1972) pp. 149–191 Uspekhi Mat. Nauk , 27 : 6 (1972) pp. 139–180
[7]	L.N. Shevrin, A.I. Ovsyannikov, "Semigroups and their subsemigroup lattices" Semigroup Forum , 27 (1983) pp. 1–154
[8]	L.N. Shevrin, A.I. Ovsyannikov, "Semigroups and their subsemigroup lattices" , 1–2 , Sverdlovsk (1990–1991) (In Russian)

Comments

The simultaneous representation of a prescribed algebraic lattice $ S $ of subalgebras, a prescribed algebraic lattice $ C $ of congruence relations, and a prescribed group $ G $ of automorphisms is treated in [a1].

References

[a1]	W.A. Lampe, "The independence of certain related structures of a universal algebra I-IV" Algebra Univ. , 2 (1972) pp. 99–112; 270–283; 286–295; 296–302