Lambda calculus was introduced around 1930 by Alonzo Church as … Proof: The variety of Boolean algebras is congruence … elements at work Theorem 8Foreveryr.e. lambda theory T … What is lambda calculus?-A theory of functions-The name of a function contains a description of the function as a program-Untypedworld: every element in lambda calculus is contemporaneously Function A possible argument fora function A possible result of the application of a function to an argument-No Partiality: every function can be applied to any …

Some negative algebraic results Theorem2 (Lusin-S. 2004) Every nontrivial lattice identity fails in the con- gruencelatticeofa suitable LAA (CA). Conclusion: We cannot apply thirty years of Universal Algebra to LAA (CA)! Lambda calculus was introduced around 1930 by Alonzo Church as part of a foundational formalism of mathematics and logic based on functions as primitive. After some years this formalism was shown inconsistent. Why? Theorem 3Classic logic is inconsistent with combinatory logic. Proof: The variety of Boolean algebras is congruence permutable. Plotkin andSimpsonhave shown that the Malcevconditions for congruence per- mutability are inconsistent with combinatory logic. Theorem 4Theimplication fragment of classic logic is inconsistent with combinatory logic. Proof: An implication algebra is 3-permutable. Plotkinand Selingerhave shown that the Malcev conditions for congruence 3-permutabilityareincon- sistentwith combinatory logic. We should be pessimistic! ….. Boolean algebras for-calculus •Let Abeanyalgebra. There exists a bijective correspondence between: -Pairs (,0) of complementary factor congruences : \0= 0 =r - Factorizations A=A/?A/0. - Decomposition operations f: A?A! A defined by f (x,y) =uixu0 y. •Lett a ( b (a)) and f a ( b (b)). (tx) y=x; (fx) y=y. (The least reflexive compatible relation on the term algebra/includ- ingt=fistrivial) •Wehaveforapair (,0) of complementary factor congruences: te 0 f) (tx) y (ex) y0 (fx) y) x (ex) y0 y. f (x,y) = (ex) y

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