Non-commutative residuated lattices
R. P. Dilworth · Transactions of the American Mathematical Society · 1939
f Lattices with a commutative multiplication have been investigated by Professor Morgan Ward and the author in a previous paper ).a>b.If © has a unit element u, the elements covered by u are called divisorfree elements of ©.If © has a null element it will be denoted by z.© is said to satisfy the ascending chain condition if every chain ai c a2 c a3 c • • ■ has only a finite number of distinct elements.Similarly if every descending chain ai d a2 s a3 d • • • has only a finite number of distinct elements, © is said to satisfy the descending chain condition.© is called archimedian if both the ascending and descending chain conditions hold.The direct product (Birkhoff [l]) of lattices Si, 22, ■ ■ ■ , 8" is defined to be the set of vectors a = {a-i, a2, • ■ ■ , a"}, a i e S< with division defined by a s & if and only if a,-d o,-.Union and crosscut are given by (a, 6) = {(a\, bi), ■ • ■ , (a", o") }, [a, b] = {[ai, h], • • • , [a", &"]}.2. Multiplication.A one-valued, binary operation xy is called a multiplication over © if the following postulates are satisfied : Mi. ab lies in © whenever a and b lie in ©.M2. a = b implies ac = bc, ca = cb.M3. a(b, c) = (ao, ac), (a, o)c = (ac, be).M4. a(6c) = (aö)c.From M2 and M3 we have (2.1) a o o implies ac s be and caocb; (2.2) [ao, ac] safe, c], [ac, be] d [a, b]c.If in addition to Mi-M4, postulate M6 below is satisfied, © is said to be a left ideal lattice.M5. asoa.In a similar manner if My is satisfied, © is said to be a right ideal lattice. MS'. a s ai».If a lattice is both a left and right ideal lattice, it is called a two-sided ideal lattice, or simply ideal lattice.Consider a lattice with unit element u over which a multiplication satisfying Mi-M4 is defined and for which M6 holds.Mg. ua = au = a.Then by M3, M5 and M6-hold so that © is an ideal lattice.A lattice with unit element in which Mo holds we call an ideal lattice with unit.© is said to be commutative if it satisfies M 7.M7. ab = ba.3. Residuation.Consider now an ideal lattice © in which the ascending * This condition may be replaced by the weaker condition that every set 5 of elements of © have a union u(S) and that u(S)c=u{Sc).t The symbol -» indicates formal implication.% As in the previous case this condition may be weakened.