By Raymond M. Smullyan

Characters from *Alice's Adventures in Wonderland* and *Through the Looking-Glass* populate those 88 exciting puzzles. Mathematician Raymond Smullyan re-creates the spirit of Lewis Carroll's writings in puzzles related to be aware play, common sense and metalogic, and philosophical paradoxes. demanding situations diversity from effortless to tricky and include ideas, plus 60 fascinating illustrations. "An creative book." — *Boston Globe*.

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Belluce, A. DiNola, P. Hajek, U. Hahle, S. Gottwald, D. Mundici, J. Pavelka, and others. A lot of results, especially those concerning residuated lattices, are taken from the book [ 51]. 24 MATHEMATICAL PRINCIPLES OF FUZZY LOGIC Residuated lattices. 7 A residuated lattice is an algebra DEFINITION £, = (L, V, A, ®, -t, 0,1). e. a ® b ::; c iff a::; b -t c. 15) We see that in comparison with the ordinary lattice, this structure is enriched by a special couple (®, -t) of operations called the adjoint couple.

16 Let £'1 = (L 1,EEl1,Q91,""1,1 1,01) and £'2 = (L 2,EEl2,Q92,""2,12,02) be MValgebras. e. it fulfils the equalities = h(x) 1812 h(y), h(x EEl1 y) = h(x) EEl2 h(y), h(""1X) = ""2h(X), h(Od = O2 , h(11) = 1 2 . h(x 1811 y) If h is a surjection, injection or bijection then we call it epimorphism, monomorphism or isomorphism, respectively. 17 Let {Aj I j E J} be a system of algebras of the same type where J is some index set. Their direct product is an algebra A whose support is the Cartesian product A = ITjEJ Aj with projections Pj : A ----+ A j , and the operations on A are defined so that if 9 is an n-ary operation on A j , j E J then for every a1, ...

8 is an MV- algebra. 39) =1- o a. 9 on page 25 is an MV-algebra. e. ,a = 1- a. +b- 1), for all a, bED. 10. In particular, it is an MV-subalgebra of the Lukasiewicz algebra £'L. 29). £, is locally finite if to every a :f. 1 there is n E N such that an = 0, or equivalently, to every a :f. 0 there is n E N such that na = 1. ) DEFINITION Let £, be an For example, Lukasiewicz MV-algebra £'L is locally finite. In the sequel, suppose that we are given some MV-algebra £'. 40) and verify the properties of partial ordering.