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By Solomon Lefschetz

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Let (L; A, V,-+,0) be a Heyting algebm with 0 i= 0* o and define x V y = (x V y)U, 0 X 0 -+ Y = XU V y. Then: (i) (R(L); A, V,~, 0) is a Boolean algebm with x' = x ~ 0 and 1 = 0* j (ii) the map cp(x) = XU is a surjective Heyting homomorphism cp : L R(L) j (iii) B(L) is a sub-bounded-lattice of R(L) . 7. 1. Consider the Boolean algebra (R(L); A, V,* ,0,1). 5), while z" = x* V0 = x ~ O. This completes the proof of (i). Besides, cp is a psudocomplemented-lattice homomorphism and (iii) holds. 13) imply cp(x -+ y) = (x -+ y)U = (XU A y*)* = (x* Vy)U = cp(x* V y) = cp(x*) Vcp(y) = (cp(x))* Vcp(y) = cp(x) ~ cp(y) .

18). 22) implies (x - y)* ~ v', therefore (x - y)* ~ x** r; y*. 9). 4. A lattice which is both a Heyting algebra and a Brouwerian algebra is called a double Heyting algebm or a bi-Brouuierian lattice (the latter 0 term was suggested by Beazer [1974a]) . 2 Classes of (relatively) (pseudo)complemented lattices 41 As already remarked, a bounded chain is a double Heyting algebra. 31') . 31') (y - x)+ = x++ Vy+ . 8. (Beazer [1974a]) . In a double Heyting algebm the operations . 32) x . 33) x=y{=}x·y=l {=}x+y=O.

By algebraic induction. The starting point is cp(XjA(' .. ,aj, . )) = cp(aj) = XjB( .. , cp(aj ), .. 15'): cp( (FiWl . . Wn(i»)A (. . ,aj , . )) = CP(Ji(WIA(. , aj, = gi(cp(WlA (. , aj , ), )), = gi(WIB(. ), ,Wn(i)A(. . , aj, ,cp(Wn(i)A( , Wn( i)B( = (FiWl . ,Wn(i»)B(... ,cp(aj), ))) ,aj, ))) , cp(aj), )) ). 1. If A is a subalgebra of B then WA(V) = WB(W) for every v E AX, where t: is the inclusion z : A - ) B. In other words, wBIAX = WA . 2. If A is a subalgebra of B and every polynomial of B is uniquely determined by its restriction on AX, then the identities of B coincide with those of A.

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Differential Equations: Geometric Theory, 2nd ed by Solomon Lefschetz


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