Difference between revisions of "Algebraic lattice"
From Encyclopedia of Mathematics
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A lattice each element of which is the union (i.e. the least upper bound) of some set of compact elements (cf. [[Compact lattice element|Compact lattice element]]). A lattice is isomorphic to the lattice of all subalgebras of some [[Universal algebra|universal algebra]] if and only if it is both complete and algebraic. These conditions are also necessary and sufficient for the lattice to be isomorphic to the congruence lattice of some universal algebra (the Grätzer–Schmidt theorem). In both cases it is assumed that the arity of the operations of the universal algebra is finite. | A lattice each element of which is the union (i.e. the least upper bound) of some set of compact elements (cf. [[Compact lattice element|Compact lattice element]]). A lattice is isomorphic to the lattice of all subalgebras of some [[Universal algebra|universal algebra]] if and only if it is both complete and algebraic. These conditions are also necessary and sufficient for the lattice to be isomorphic to the congruence lattice of some universal algebra (the Grätzer–Schmidt theorem). In both cases it is assumed that the arity of the operations of the universal algebra is finite. | ||
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Gierz, | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Birkhoff, "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc. (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G. Grätzer, "General lattice theory" , Birkhäuser (1978) (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)</TD></TR> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.V. Mislove, D.S. Scott, "A compendium of continuous lattices" , Springer (1980) {{ISBN|3-540-10111-X}} {{MR|0614752}} {{ZBL|0452.06001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Grätzer, E.T. Schmidt, "Characterizations of congruence lattices of abstract algebras" ''Acta. Sci. Math. (Szeged)'' , '''24''' (1963) pp. 34–59</TD></TR></table> | ||
Latest revision as of 16:58, 25 November 2023
compactly-generated lattice
A lattice each element of which is the union (i.e. the least upper bound) of some set of compact elements (cf. Compact lattice element). A lattice is isomorphic to the lattice of all subalgebras of some universal algebra if and only if it is both complete and algebraic. These conditions are also necessary and sufficient for the lattice to be isomorphic to the congruence lattice of some universal algebra (the Grätzer–Schmidt theorem). In both cases it is assumed that the arity of the operations of the universal algebra is finite.
Comments
Algebraic lattices are an important special case of continuous lattices, for which see [a1]. The original reference to the Grätzer–Schmidt theorem is [a2].
References
| [1] | G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967) |
| [2] | G. Grätzer, "General lattice theory" , Birkhäuser (1978) (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978) |
| [a1] | G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.V. Mislove, D.S. Scott, "A compendium of continuous lattices" , Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001 |
| [a2] | G. Grätzer, E.T. Schmidt, "Characterizations of congruence lattices of abstract algebras" Acta. Sci. Math. (Szeged) , 24 (1963) pp. 34–59 |
How to Cite This Entry:
Algebraic lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_lattice&oldid=17011
Algebraic lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_lattice&oldid=17011
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article