Algebras inspired by the equivalential calculus

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1 Algebras inspired by the equivalential calculus Wies law A. DUDEK July 8, 1998 Abstract The equivalential calculus was introduced by Leśniewski in his paper on the protothetic [8]. Such logical system maybe defined by the rule of detachment and any one of the formulae EEpqEErqEpr, EEpqEEprErq, EEpqEErpEqr. In this note we give a characterization of two classes of groupoids connected with this calculus. From obtained characterization follows that the formulae EEpqEErpErq, EEpqEErqErp, EEpqEEqrEpr and EEpqEEprEqr are not a sole axioms of the equivalential calculus. 1. Introduction The Polish logician S. Leśniewski, and some of his collaborators were particularly interested in the extension, by the use of quantifiers and functorial variables, of versions of the propositional calculus in which the only undefined truth functor is E, If and only if. Just as there is a restricted segment of the propositional calculus in which no symbols are needed but propositional variables and the operator C, so there is another restricted segment in which no symbols are needed but propositional variables and the operator E. (Note that in the notion of Lukasiewicz Cpq denotes If p then q. This functor C is called the implication by p of q, or a conditional proposition. Thus the above Leśniewski calculus may be called the equivalential calculus.) This calculus was introduced by Leśniewski in his paper on the protothetic [8]. Among the more obvious theses occurring in it would be EEpqE qp, Epp and EEpqEEqrEpr, or in the Peano-Russellian symbolism, p q..q p, p p and p q : : q r..p r. Leśniewski and Lukasiewicz have shown that any formula constructed by equating equivalences is a thesis if all propositional variables in it occurs an even number of times [7]. AMS Subject Classification (2000): 03 G 25, 20 N 02. Key words and phrases: equialential calculus, protothetic, groupoid. 1

2 Moreover, J. Lukasiewicz proved in 1939 (cf. [11]) that any one of the formulae EEpqEErqEpr, EEpqEEprErq and EEpqEErpEqr will serve as a sole axiom for the system. Some other axiom systems are given by Y.Arai, K.Iséki and S.Tanaka in [1], [2], [6], [12] and [13]. 2. Algebras and logics By an algebra (i.e. a groupoid) we mean a non-empty set G together with a binary multiplication (denoted by juxtaposition) and a some distinguished element 0. Such an algebra is denoted by (G,, 0). Each such algebra will have certain equality axioms (including x = x) and the rule of substitution of equality as well as perhaps some other rules. Many of such algebras were inspired by some logical systems. This inspiration is illustrated by the similarities between the names. In many cases, the connection between such algebras and their corresponding logics is much stronger. In this case one can give a translation procedure which translates all well formed formulas and all theorems of a given logic L into terms and theorems of the corresponding algebra (cf. [10]). In some cases one can give also an inverse translation procedure which translates all terms and all theorems of this algebra into well formed formulas and theorems of a given logic L (cf. [5]). In this case we say that the logic L and its corresponding algebra are isomorphic. Nevertheless the study of algebras motivated by known logics is interesting and very useful for corresponding logics, also in the case when these structures are not isomorphic. In this note we describe algebras which correspond to the equivalential calculus introduced by S.Leśniewski. This calculus is formulated as follows: Let M be an arbitrary abstract non-empty set with the only one undefined truth functor as a primitive notion. If the structure M = (M, ) satisfies the following axioms: a) p r..q p :.r q, b) p.q r : : p q. r, then it is called the equivalential calculus. In the equvalential calculus, we use the rule of usual substitution and the rule of detachment: α and α β imply β. The above equivalential calculus may be translated into an algebra (G,, 0) in the following way: (i) (ii) if x, y,... are variables of the equivalential calculus M and if p, q,... are well formed formulas of M then x = x and (p q) = q p, if p q is a logical statement then q p = 0 is the translated algebraic statement. 2

3 In this translation the above axioms of the equivalential calculus are translated into a ) (q r) ((p q) (r p)) = 0, b ) (r (q p)) ((r q) p) = 0. As was mentioned above, in the sequel a multiplication will be denoted by juxtaposition. Dots we use only to avoid repetitions of brackets. Using this convention the above axioms will be written as qr (pq rp) = 0 and (r qp)(rq p) = 0. The rule of detachment may be translated as: α = 0 and βα = 0 imply β = 0, i.e. β0 = 0 imply β = E-groupoids A non-empty set G with a binary operation denoted by juxtaposition and a distinguished element 0 is called an E-groupoid if the following conditions hold: (1) xy (zx yz) = 0, (2) (x yz)(xy z) = 0, (3) x0 = 0 implies x = 0. It is clear that such E-groupoid corresponds to the eqivalential calculus. As it is proved in [3] and [4], every such groupoid satisfies the following conditions: (4) 0x = 0 implies x = 0, (5) xy z = 0 iff x yz = 0, (6) x 0x = 0, (7) x0 x = 0, (8) xx = 0, (9) xy yx = 0, (10) xy = 0 iff yx = 0, (11) (xy z)(x yz) = 0, (12) xy y = 0 implies x = 0, (13) ((xy z)x) yz = 0, (14) ((xy (zx yz))((zx y)z)) xy = 0. Basing on the above conditions one can proved (cf. [4], Proposition 2) that in the definition of an E-groupoid the axiom (2) may be replaced by (11). Moreover, the following theorem is true. Theorem 1. The class of all E-groupoids is a uniquely defined by (3), (9) and (11). 2 3

4 Now we give the another characterization of E-groupoids. Theorem 2. A groupoid (G,, 0) satisfying (3) is an E-groupoid iff it satisfies one of the following identities: (15) (xy zx) yz = 0, (16) (xy zx) zy = 0, (17) (xy yz) xz = 0. Proof. Assume that a groupoid (G,, 0) satisfies (3). (E) (15). If (G,, 0) is an E-groupoid, then it satisfies (15), because it satisfies (1) and (5). (15) (16). If (G,, 0) satisfies (3) and (15), then replacing in (15) x by u, y by xy zx, z by yz and using (15) with (3) we obtain (18) u(xy zx) (yz u) = 0, which for x = u, y = xy, z = zx, u = yz implies (by (15) and (3)) yz ((u xy) (zx u)) = 0. This for x = yz, y = yz y gives (by (18) where x = y, y = z, z = yz) (yz y)z = 0. Putting in (18) y = yz, z = y, u = z and applying the above identity we obtain z((x yz) yx) = 0, which for y = z = xx gives (by (18)) xx = 0. Therefore from (15) follows (9), and in the consequence - (10). Thus (18) for u = zy gives zy (xy zx) = 0, which by (10), is equivalent to (16). Hence (15) implies (16). (16) (17). If (16) holds, then (yy yy) yy = 0, which together with (16) and (3) implies (x yy) (yy yy)x = 0. Since this for x = yy gives yy yy = 0 and 0 yy = 0, then this identity may be written as (x yy) 0x = 0. Replacing x by yy and applying the above identities we get 00 = 0, which together with (16) proves x0 0x = 0. Thus 0 = (yy 0)(0 yy) = (yy 0)0 = yy 0 = yy, i.e. yy = 0. Hence as a consequence of (16) we obtain (9). Moreover, putting in (16) x = xz, z = zx and using (9) we get (xz y)0 (zx y) = 0 which shows that zx y = 0 implies xz y = 0. Hence from (16) follows (zx xy) zy = 0. This identity is obviously equivalent to (17). Thus every groupoid satisfying (3) and (16) satisfies also (17). (17) (E). Assume that a groupoid (G,, 0) satisfies (3) and (17). We prove that it satisfies also (9) and (11) (cf. Theorem 1). To prove this fact, observe first that this groupoid satisfies (8). Indeed, replacing in (17) x by xu uz, z by xz and applying (17) together with (3) we get (19) (xu uz)y (y xz) = 0, 4

5 which for x = y, y = (xu uz)y and z = xz gives (by (19) and (3)) (yu (u xz)) (xu uz)y = 0. For y = xz from this identity follows (by (17)) (20) (xz u)(u xz) = 0, which for u = xy yz gives (by (17) and (3)) (21) xz (xy yz) = 0. Applying (21) and (3) to (19), where x = xy, y = xz, z = yz, we have ((xy u)(u yz)) xz = 0. Further, putting in (20) u = (xy u)(u yz) and using the last identity, we obtain xz ((xy u)(u yz)) = 0, which for y = z = x gives (by (20)) xx = 0. This proves (8). As a simple consequence of (8), (17) and (3) we obtain (9). This implies (10). To prove (11) observe that replacing in (17) y by yz and z by zy, we obtain ((x yz)(yz zy)) (x zy) = 0, which by (9) and (3) shows that (22) x zy = 0 implies x yz = 0. Similarly, replacing in (17) x by xy, y by z, z by yx, applying (3), (9) and (22) we get (xy z)(yx z) = 0. Now, putting x = xu z, z = ux z in (17), and using the last identity we have (xu z)y y(ux z) = 0, which shows that (23) y(ux z) = 0 implies (xu z)y = 0. Since (21) for y = xy and z = y xy has the form x(y xy) [(x xy)(xy (y xy))] = 0, then applying (21) with z = xy to the last square bracket of the above identity we obtain (x(y xy))0 = 0, which by (3) and (22) gives x(xy y) = 0. This together with (23), (10) and (22) implies (24) x(y yx) = 0. Since using (17), (22), (10), (22) and (23) to (17) we have 0 = ((yx z) zy)(yx y) = ((yx z) zy)(y yx) = (y yx)((yx z) zy) = (y yx)(zy (yx z)) = (yz (yx z))(y yx), then putting in (17) x = yz (yx z), y = x, z = y yx we get [((yz (yx z))x)0] [(yz (yx z))(y yx)] = 0, 5

6 which implies (25) (yz (yx z))x = 0. This for x = 0 gives yz (y0 z) = 0. Replacing in this identity y by yz x, z by yx z, 0 by (yz (yx z))x (cf. (25)) we obtain [(yz x)(yx z)] [((yz x)((yz (yx z))x))(yx z)] = 0. Since by (25) (where y = yz, z = x, x = yx z) the last square bracket of the above identity is equal to 0, then this identity implies (yz x)(yx z) = 0, which by (23) and (22) gives (11). This proves that (3) and (17) imply (9) and (11). Hence (Theorem 1) this groupoid is an E-groupoid. The proof is complete. 2 Corollary. A groupoid (G,, 0) satisfying (3) and (9) is an E-groupoid iff it satisfies one of the following identities: (26) (xy zy) xz = 0, (27) (xy zy) zx = 0, (28) (xy xz) zy = 0, (29) (xy xz) yz = 0, (30) (xy yz) zx = 0. Proof. It is not difficult to see that the above identities are a consequence of (5), (10), (22) and identities from Theorem 2. Thus its holds in any E-groupoid. To prove the converse observe that (3) and (9) imply (4) and (10). This together with (26), where x = xz, z = zx, gives (27). Similarly (28) gives (29). Since (27) for x = xz, z = zx implies (xz y)(zx y) = 0, then replacing in (27) x by xy zx, y by zy, z by yx zx and using the above identity we obtain (16) which proves that a groupoid with (3), (9) and (27) is an E-groupoid. Now let (3), (9) and (29) hold. Putting x = xy, y = zx, z = xz in (29) and using (9) we obtain (xy zx)(xy xz) = 0. Since yz (xy xz) = 0 by (10), then replacing in (29) x by yz, y by xy zx, z by xy xz and applying the above identities and (10) we get (15). Thus groupoid satisfying (3), (9) and (29) is an E-groupoid. If (3), (9) and (30) hold then from (30) for x = zx, z = xz follows (zx y)(y xz) = 0, which by (10) shows that y xz = 0 iff y zx = 0. This proves that (30) and (17) are equivalent, which completes our proof. 2 Note that there are groupoids in which (3) and one of the following identities (26) - (29) hold, but (9) is not satisfied. As an example we may consider the groupoid (G,, 0) defined on an arbitrary commutative group (G, +, 0) of the exponent k > 2, where xy = x y. This groupoid satisfies (3), (26) and (28) 6

7 but not satisfies (9). Putting xy = y x we obtain the groupoid satisfying (3), (27) and (29) but not (9). From the above examples follows that any of the formulae EEpqEErpErq, EEpqEErqErp, EEpqEEqrEpr, EEpqEEprEqr is not a sole axiom for the Leśniewski s equivalential calculus. The problem of the formula EEpqEEqrErp (which may be translated into (30)) is open. Note that in general E-groupoids are not commutative (cf. [4]). Moreover, in general, commutative E-groupoids are not medial, i.e. in E-groupoids the commutativity not implies xy zu = xz yu. Indeed, from Theorem 1 follows that the set G with a distinguished element 0 and at least two elements x, y 0 is a commutative E-groupoid with respect to the operation defined by 0 x = x 0 = x and x y = 0 for all x 0, y 0. In this E-groupoid we have (0 x) (y x) = x (0 y) (x x) = y for all x, y G, which proves that this E-groupoid is not medial. Theorem 3. Any E-groupoid is weak medial, i.e. in any E-groupoid xy zu = 0 implies xz yu = 0. Proof. Indeed, applying (5) and (22) to (2) we obtain x(xy z) yz = 0. This for y = yz u and z = uy z gives x(x(yz u) (uy z)) = 0, because (yz u)(uy z) = 0 by (2), (10) and (21). Thus x(yz u) = 0 (by (22) and (5)) implies x(uy z) = 0. Now, using (5), (22) and the last implication we have 0 = xy zu = xy uz = x(y uz) = x(uz y) = x(yu z) = x(z yu) = xz yu, which completes the proof. 2 Let xρy iff xy = 0. From (8) and (10) immediately follows that this relation is reflexive and symmetric. By (22) and Theorem 3 it is also transitive. Thus ρ is an equivalence relation. Moreover Theorem 3 proves that it is a congruence too. Since x and y are in the same equivalence class C z iff xy = 0, then C 0 = {0} and it is not difficult to verify that the set G/ρ of all equivalence classes is a Boolean group with respect to the multiplication defined by C x C y = C xy. Hence the multiplication table of an E-groupoid has a special form. Namely, an element 0 occupies the whole main diagonal of such table and all places in which occurs 0 are symmetric with respect to this diagonal. Thus there exists only one (up to isomorphism) two-elements E-groupoid. Obviously it is a Boolean group. All three-elements E-groupoids have multiplication tables described in [4]. A similar table has a four-elements E-groupoid which is not a Boolean group. (From Proposition 7 in [4] follows that an E-groupoid (G,, 0) is a Boolean group iff xy = 0 implies x = y.) 7

8 4. E 0 -groupoids In the set of all E-groupoids we select these groupoids in which the identity x0 = x holds, i.e. E-groupoids in which 0 is a right neutral element. Such groupoids will be called E 0 groupoids. It is clear that the class of all E 0 -groupoids forms a proper variety contained in the quasivariety of all E-groupoids. But in an E-groupoid (G, ) the set {x G : x0 = x} is not a subgroupoid. Also 0x = x is not true, in general. Theorem 4. A groupoid (G,, 0) satisfying x0 = x is an E 0 -groupoid iff it satisfies one of the identities: (15), (16), (17), (27), (29), (30). Proof. Let x0 = x holds in (G,, 0). From Theorem 2 follows that it is E -groupoid iff it satisfies one of the identities (15), (16), (17). If it satisfies (27) then it satisfies also (9), which follows from (27) for y = 0. Thus by Corollary (G,, 0) is an E 0 -groupoid. Since for y = z = 0 (29) gives (8), then (29) for x = z implies (9). Thus (G,, 0) with (29) is an E 0 -groupoid. Now, if (30) holds then x 0x = 0. Moreover, putting x = zx, z = xy yx in (30) we obtain (zx y) y(xy yz) = 0. But this for z = x, y = 0 gives xx = 0, which together with (30) implies (9). Hence also in this case (G,, 0) is an E 0 -groupoid. 2 References [1] Y.Arai: On axiom systems of propositional calculi. XVII, Proc. Japan Acad. 42 (1966), [2] Y.Arai, S.Tanaka: On axiom systems of propositional calculi. XIX, Proc. Japan Acad. 42 (1966), [3] W.A.Dudek: Algebras connected with the equivalential calculus, Math. Montisnigri 4 (1995), [4] W.A.Dudek: Algebras motivated by the equivalential calculus, Rivista Mat. Pura et Appl. 17 (1996), [5] H.G.Forder, J.A.Kalman: Implication in equational logic, Math. Gazette 46 (1962), [6] K.Iséki: On axiom systems of propositional calculi. XV, Proc. Japan Acad. 42 (1966), [7] Z.Jordan: The development of mathematical logic and logical positivism in Poland between the two wars, being Polish Science and Learning No 6 (Oxford University Press, 1945). 8

9 [8] S.Leśniewski: Grundrüge eines neuen Systems der Grundlagen der Mathematik, Fund. Math. 14 (1929), [9] A.N.Prior: Formal logic, 2nd ed. Oxford [10] H.Rasiowa: An algebraic approch to non-classical logics, North-Holland, Amsterdam [11] B.Sobociński: An investigation of protothetic, Éditions de l Institut d Études Polonaises en Belgique (Brussels, 1949). [12] S.Tanaka: On axiom systems of propositional calculi. XVIII, Proc. Japan Acad. 42 (1966), [13] S.Tanaka: On axiom systems of propositional calculi. XX, Proc. Japan Acad. 42 (1966), Institute of Mathematics, Technical University of Wroc law, Wybrzeże Wyspiańskiego 27, Wroc law, Poland dudek@im.pwr.wroc.pl 9