Lattice with complements

2020 Mathematics Subject Classification: Primary: 06C15 [MSN][ZBL]

complemented lattice

A lattice $L$ with a zero 0 and a unit 1 in which for any element $a$ there is an element $b$ (called a complement of the element $a$) such that $a\lor b=1$ and $a\land b=0$. If for any $a,b\in L$ with $a\leq b$ the interval $[a,b]$ is a complemented lattice, then $L$ is called a relatively complemented lattice. Each complemented modular lattice is a relatively complemented lattice. A lattice $L$ with a zero 0 is called:

a) a partially complemented lattice if each of its intervals of the form $[0,a]$, $a\in L$, is a complemented lattice;

b) a weakly complemented lattice if for any $a,b\in L$ with $b\nleq a$ there is an element $c\in L$ such that $a\land c=0$ and $b\land c\neq0$;

c) a semi-complemented lattice if for any $a\in L$, $a\neq1$, there is an element $b\in L$, $b\neq0$, such that $a\land b=0$;

d) a pseudo-complemented lattice if for any $a\in L$ there is an element $a^*$ such that $a\land x=0$ if and only if $x\leq a^*$; and

e) a quasi-complemented lattice if for any $x\in L$ there is an element $y\in L$ such that $x\land y=0$ and $x\lor y$ is a dense element.

Ortho-complemented lattices also play an important role (see Orthomodular lattice). See [4] for the relation between the various types of complements in lattices.

References

[1]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[2]	L.A. Skornyakov, "Elements of lattice theory" , A. Hilger (1977) (Translated from Russian)
[3]	L.A. Skornyakov, "Complemented modular lattices and regular rings" , Oliver & Boyd (1964) (Translated from Russian)
[4]	P.A. Grillet, J.C. Varlet, "Complementedness conditions in lattices" Bull Soc. Roy. Sci. Liège , 36 : 11–12 (1967) pp. 628–642

Comments

In a distributive lattice, each element has at most one complement; conversely, a lattice in which each element has at most one relative complement in each interval in which it lies must be distributive.

A Boolean lattice is a complemented distributive lattice.

References

[a1]	L. Beran, "Orthomodular lattices" , Reidel (1985)
[a2]	G. Grätzer, "Lattice theory" , Freeman (1971)
[a3]	M.L. Dubreil-Jacotin, L. Lesieur, R. Croiset, "Leçons sur la théorie des treillis" , Gauthier-Villars (1953)
[b1]	B. A. Davey, H. A. Priestley, Introduction to lattices and order, 2nd ed. Cambridge University Press (2002) ISBN 978-0-521-78451-1 Zbl 1002.06001