ALGEBRAIC STRUCTURES OF GENERALIZED MANY-VALUED LOGICS (SEMINAR NOTES) Costas A. Drossos Department of Mathematics University of Patras GR-26110 Patras, GREECE First Draft (These are notes from a seminar given in the Department of Mathematics, University of Patras 1996–97) 2 Residuated Monoids CONTENTS 1 POSETS 1 1.1 Preordered Sets .................................. 1 1.2 Order Ideals and Filters ............................. 4 1.3 Mappings Between Posets ............................ 5 2 ADJOINTS FOR PREORDERS 11 2.1 Adjoints ...................................... 11 2.2 Triangle Inequalities and Basic Properties of Adjunctions. ...................... 13 2.3 Adjoint Functor Theorem ............................ 14 2.4 Closure operators (monads) on preorders .................... 18 2.5 The Adjoint Lifting Theorem. .......................... 24 3 cirl-MONOIDS 27 3.1 Introduction .................................... 27 3.2 The Qualitative Case: Heyting Algebras. .................... 29 3.3 The Quantitative Case:cirl-monoids ....................... 30 3.4 Monoid actions. .................................. 33 3.5 Integrality. ..................................... 43 i 0 Residuated Monoids 3.6 Algebraic strong de Morgan’s law ........................ 44 3.7 Divisibility. .................................... 48 3.8 Involutive Negation and Girard monoids. ................... 53 3.9 Square roots. ................................... 55 CHAPTER 1 POSETS 1.1 Preordered Sets In these notes we shall need many implication symbols. To fix notation, we shall use ² ‘ ) ’ for ‘implies’; ² ‘ , ’ for ‘if and only if’; ² ‘ =^) ’ for qualitative (lattice) implication; ² ‘ (_=’ for qualitative (lattice) co- implication; ² ‘ =¯) ’ for quantitative (monoidal) implication. ² ‘ (©=’ for quantitative (monoidal) co-implication. ² ‘ ¡¡! ’, for function arrow. 1.1.1 Definition. A preordered set is a pair hP; ·i, consisting of a set P and a binary relation “·” on P such that: (i) Reflexivity: (8p 2 P )[p · p] (ii) Transitivity: (8p; q; r 2 P )[(p · q) ^ (q · r) ) (p · r)] A preordered set it is not required to satisfy the antisymmetric property: (iii) Antisymmetric: (8p; q 2 P )[(p · q) ^ (q · p) ) (p = q)] 1 2 Residuated Monoids If (iii) is satisfied by P then it is called a poset. Two elements p; q 2 P are said to be comparable if p · q or q · p (dichotomy property), otherwise we say that p and q are incomparable, denoted by p k q.A totally ordered set or a chain is a poset in which every two elements are comparable. Often we shall present the properties of a poset in a Gentzen-like style: (reflexivity) x · x x · y y · z (transitivity) x · z x · y y · x (antisymmetric) x = y One line means that the statements above the line imply the statement below, two lines mean that the above and below are logically equivalent. 1.1.2 Remark. Given any preordered set P , there is a naturally associated poset, denoted by P= ´ obtained from P , as a quotient structure, where “´” is defined as (8p; q 2 P )[(p ´ q) iff (p · q) ^ (q · p)]: Similarly for the order on P= ´. Because of the above and the fact that in Category Theory “equality” means “isomorphic”, preordered sets are more naturally adapted to a categorical treatment of order. 1.1.3 Definition. 1) Given a poset P = hP; ·i we may define the dual of opposite of P, denoted by Pop = hP; ¸i. The duality principle for posets allows us to dualize any theorem by interchanging · with ¸. Y 2) The direct product of a family of posets hPi; ·iii2I is a poset h Pi; ·i, where “·” is i2I defined componentwise, i.e. (xi)i2I · (yi)i2I iff xi ·i yi for all i 2 I. If the index set I is also endowedY with a total order, then the lexicographic product of the above family is a poset h Pi; ·lexi where “·lex” is defined as i2I (xi)i2I ·lex (yi)i2I iff (9i 2 I)[(xi · yi) ^ (8k < i)[xk = yk]] If in addition to the above, we have that Pi[\ Pj = ;, whenever i 6= j, then we may define the ordinal sum of the above family, as: h Pi; ·i, where “·” is defined as i2I x · y iff (9i 2 I)[x ·i y] or (9i; j 2 I)[i < j and x 2 Pi; & y 2 Pj]: . 1: POSETS 3 1.1.4 Definition. (i) Let hP; ·i be a poset, and S ⊆ P . Then an element u 2 P is called an upper bound of S iff for every p 2 S, we have p · u. Dually we may define lower bound of S. (ii) A poset hP; ·i is called directed upwards iff every pair of elements p; q 2 P has an upper bound. Similarly, hP; ·i is called directed downwards iff every pair of elements p; q 2 P has a lower bound. If hP; ·i is directed above and below it is called simply a directed set. 1.1.5 Definition. Let hP; ·i be a poset. If for every pair of elements p; q 2 P , the set of upper bounds of fp; qg has a least element r, then r is called the least upper bound (join) of p and q and is denoted as r = p _ q. The greatest lower bound (meet), denoted p ^ q is defined dually. More precisely, let us denote by Su;S ⊆ P , the set of all upper bounds of S (upper cone), and S` the set of all lower bounds (lower cone), i.e. Su := fp 2 P j(8s 2 S)[s · p]g and S` := fp 2 P j(8s 2 S)[s ¸ p]g Now we define: _ _ sup S ´ S ´ s := inf Su s2S ^ ^ inf S ´ S ´ s := sup S`: s2S In particular we have: _ _ ^ ^ ; ´ p = 0 and ; ´ p = 1 p2; p2; the bottom and top elements if they exist. If for every pair of elements p; q 2 P , the joint p _ q exists in P then hP; ·i is called an upper semilattice. Dually, if p ^ q exists for every pair p; q 2 P , then hP; ·i is called a lower semilattice. If hP; ·i is both an upper and a lower semilattice, then it is called a lattice. Compaire with the concept of ‘directed set’. 1.1.6 Example. Posets. 1. Any set X with the discrete order: x · y iff x = y; 2. N; Z; Q and R with the usual order; 3. 2 ´ h2; ·i where 2 = f0; 1g and 0 · 1; 4 Residuated Monoids 4. if hP; ·i is a poset and X an arbitrary set, then the set P X := ff j f : X ¡¡! P g, endowed with the pointwise order, f · g iff (8x 2 X)[f(x) · g(x)] is a poset. 5. For any set X, hP (X); ⊆i is a poset. If ¿ ⊆ P (X) is a topology on X, then h¿; ⊆i with the restriction order is also a poset. 6. The specialization order between points in a topological space, x · y iff (8G 2 ¿)[x 2 G ) y 2 G] is in ceneral a preorder. The specialization order is antisymmetric iff ¿ is a T0 space. If ¿ is a T1 then the specialization order is a discrete order. 7. Formulae of a propositional logic form a preorder under provability `. 8. Let G = hG0;G1i be an oriented multigraph with loops, see [1]. A subgraph X = hX0;X1i is defined as: X0 ⊆ G0;X1 ⊆ G1 such that for every edge in X1, its source and target are in X0, i.e. P (G) := fX = hX0;X1i j X0 ⊆ G0;X1 ⊆ G1 such that (8f 2 X1)[dom(f); cod(f) 2 X0]g: We now define on P (G) a partial order by: For X; Y 2 P (G), X ⊆ Y iff X0 ⊆ Y0 and X1 ⊆ Y1: Then hP (G); ⊆i is a poset. 1.2 Order Ideals and Filters Let hP; ·i be a poset and Q ⊆ P . 1.2.1 Definition. (i) Q is called an order ideal (or o-ideal or a down-set) iff (8p)(8q)[(p 2 Q) ^ (q · p) ) (q 2 Q)]: (ii) Dually, Q is called an order filter (or o-filter or an up-set) iff (8p)(8q)[(p 2 Q) ^ (q ¸ p) ) (q 2 Q)]: 1.2.2 Proposition. Q is a down-set iff P n Q is an up-set. 1: POSETS 5 Given an arbitrary subset Q of P and x 2 P , we define: #Q := fy 2 P j (9x 2 Q)[y · x]g (down Q) "Q := fy 2 P j (9x 2 Q)[y ¸ x]g (up Q) #x ´# fxg := fy 2 P j y · xg (Principal order-ideal) "x ´ "fxg := fy 2 P j y ¸ xg (Principal order-filter) 1.2.3 Remark. A preorder hP; ·i can be considered as a category C(P; ·): ² The objects of C(P; ·) are the elements of P ² If x; y 2 P and x · y then C(P; ·) has exactly one arrow from x to y denoted, x ¡¡! y of (x; y). Note that dom(x ¡¡! y) = x, cod(x ¡¡! y) = y. ² If x k y (x; y are not comparable) there is no arrow from x to y. ² The identity arrows of C(P; ·) are those of the form (x; x) or x ¡¡! x. The transitive property of “·” is needed to ensure the existence of composition of two arrows, so that: x ¡¡! y y ¡¡! z or (x ¡¡! y) ± (y ¡¡! z) = (x ¡¡! z): x ¡¡! z Slogan. Think of a category as a generalized preordered set. We shall take the above slogan seriously and try to see what is the special form which category concepts take in the enviroment of preordered sets. Thus, a down-set, #x = fy 2 P j y ¡¡! xg is all the arrows y ¡¡! x, for y 2 P , which is what is called a slice category P #x.