De Morgan algebra

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a De Morgan algebra (named after <a href="/facts/Augustus_De_Morgan/6fQ1KTPb">Augustus De Morgan</a>, a British mathematician and logician) is a structure A = (A, ∨, ∧, 0, 1, ¬) such that:

<ul><li>(A, ∨, ∧, 0, 1) is a <a href="/facts/Lattice_(order)/5ak6ufky">bounded</a> <a href="/facts/Distributive_lattice/97DFz1cu">distributive lattice</a>, and</li>
<li>¬ is a De Morgan involution: ¬(x ∧ y) = ¬x ∨ ¬y and ¬¬x = x. (i.e. an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a> that additionally satisfies <a href="/facts/De_Morgan%2527s_laws/dj3F8xpt">De Morgan's laws</a>)</li></ul>
In a De Morgan algebra, the laws

<ul><li>¬x ∨ x = 1 (<a href="/facts/Law_of_the_excluded_middle/tDhbugCN">law of the excluded middle</a>), and</li>
<li>¬x ∧ x = 0 (<a href="/facts/Law_of_noncontradiction/cFkRnVsu">law of noncontradiction</a>)</li></ul>
do not always hold. In the presence of the De Morgan laws, either law implies the other, and an algebra which satisfies them becomes a <a href="/facts/Boolean_algebra_(structure)/cHbo65jm">Boolean algebra</a>.
Remark: It follows that ¬(x ∨ y) = ¬x ∧ ¬y, ¬1 = 0 and ¬0 = 1 (e.g. ¬1 = ¬1 ∨ 0 = ¬1 ∨ ¬¬0 = ¬(1 ∧ ¬0) = ¬¬0 = 0). Thus ¬ is a dual <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a> of (A, ∨, ∧, 0, 1).
If the lattice is defined in terms of the order instead, i.e. (A, ≤) is a bounded partial order with a least upper bound and greatest lower bound for every pair of elements, and the meet and join operations so defined satisfy the distributive law, then the complementation can also be defined as an involutive anti-automorphism, that is, a structure A = (A, ≤, ¬) such that:

<ul><li>(A, ≤) is a <a href="/facts/Lattice_(order)/5ak6ufky">bounded</a> <a href="/facts/Distributive_lattice/97DFz1cu">distributive lattice</a>, and</li>
<li>¬¬x = x, and</li>
<li>x ≤ y → ¬y ≤ ¬x.</li></ul>
De Morgan algebras were introduced by <a href="/facts/Grigore_Moisil/6lyIqIyB">Grigore Moisil</a> around 1935, although without the restriction of having a 0 and a 1. They were then variously called quasi-boolean algebras in the <a href="/facts/Lw%25C3%25B3w%25E2%2580%2593Warsaw_school/3JTpUm5k">Polish school</a>, e.g. by <a href="/facts/Helena_Rasiowa/XVNygStk">Rasiowa</a> and also distributive i-lattices by J. A. Kalman. (i-lattice being an abbreviation for lattice with involution.) They have been further studied in the Argentinian algebraic logic school of <a href="/facts/Antonio_Monteiro_(mathematician)/PStwtS5G">Antonio Monteiro</a>.
De Morgan algebras are important for the study of the mathematical aspects of <a href="/facts/Fuzzy_logic/1PvN4Att">fuzzy logic</a>. The standard fuzzy algebra F = ([0, 1], max(x, y), min(x, y), 0, 1, 1 − x) is an example of a De Morgan algebra where the laws of excluded middle and noncontradiction do not hold.
Another example is <a href="/facts/J._Michael_Dunn/124R0W5o">Dunn</a>'s four-valued semantics for De Morgan algebra, which has the values T(rue), F(alse), B(oth), and N(either), where

<ul><li>F 
<li>F < N < T, and</li>
<li>B and N are not comparable.</li></ul>

De Morgan algebra open-in-new

De Morgan algebra