Soft Comput (2013) 17:45–47 DOI 10.1007/s00500-012-0901-x

ORIGINAL PAPER

Congruences generated by ideals of the compatibility center of lattice effect algebras Gejza Jencˇa

Published online: 24 July 2012 Ó Springer-Verlag 2012

Abstract We prove that for every lattice effect algebra, the system of all congruences generated by the prime ideals of the compatibility center separates the elements. This is a common generalization of Chang’s representation theorem from 1959 and a result of Graves and Selesnick (Colloq Math 27:21–30, 1973). Keywords Effect algebra  Lattice effect algebra  Compatibility center

1 Definitions An effect algebra is a partial algebra ðE; ; 0; 1Þ with a binary partial operation  and two nullary operations 0,1 satisfying the following conditions. (E1) (E2)

(E3) (E4)

If a  b is defined, then b  a is defined and a  b ¼ b  a: If a  b and ða  bÞ  c are defined, then b  c and a  ðb  cÞ are defined and ða  bÞ  c ¼ a ðb  cÞ: For every a 2 E there is a unique a0 2 E such that a  a0 ¼ 1: If a  1 exists, then a = 0.

Effect algebras were introduced by Foulis and Bennett (1994) in their paper. Chovanec and Koˆpka (1994) introduced an essentially equivalent structure called D-poset. Their definition is an

G. Jencˇa (&) Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak Technical University, Radlinske´ho 11, Bratislava 813 68, Slovak Republic e-mail: [email protected]

abstract algebraic version the D-poset of fuzzy sets, introduced by Koˆpka (1992). Another equivalent structure was introduced by Giuntini and Greuling (1989). We refer to Dvurecˇenskij (2000) for more information on effect algebras and related topics. In an effect algebra E, we write a B b iff there is c 2 E such that a  c ¼ b: It is easy to check that every effect algebra is cancellative, thus B is a partial order on E. In this partial order, 0 is the least and 1 is the greatest element of E. Moreover, it is possible to introduce a new partial operation ; b  a is defined iff a B b and then a  ðb  aÞ ¼ b: It can be proved that a  b is defined iff a  b0 iff b  a0 : Therefore, it is usual to denote the domain of  by \. Let E be an effect algebra. A subset I of E is called an ideal of E iff the following condition is satisfied. x; y 2 I and x ? y , x  y 2 I An ideal I of an effect algebra E is called Riesz ideal iff the following condition is satisfied: if i 2 I and i  a  b; then there exist i1 ; i2 2 I; such that i1  a; i2  b; i  i1  i2 : If E is an orthomodular lattice, then an ideal I of E is Riesz iff I is closed with respect to perspectivity; such ideals are called p-ideals or orthoideals in the theory of orthomodular lattices. In what follows, RI(E) denotes the set of all Riesz ideals in an effect algebra E. It was proved by Chevalier (2000) that RI(E) is a distributive sublattice of the lattice of all ideals. Given a Riesz ideal I of an effect algebra E, the relation *I on E is defined as follows: a*I b iff there are i; j 2 I such that i  a; j  b; ða  iÞ ¼ ðb  jÞ: When equipped with a partial  inherited from E, E/*I is then an effect algebra (see Gudder 1998 for a proof of this). Moreover, as proved by Jencˇa (2001), the class of lattice effect algebras is closed with respect to quotients modulo Riesz ideals. In an effect algebra E, a pair of elements a, b is called compatible (a $ b) iff there are a1 ; b1 ; c 2 E such that

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a1  b1  c exists and a ¼ a1  c; b ¼ b1  c: In a lattice effect algebra E, we have a $ b iff ða _ bÞ  b ¼ a  ða ^ bÞ (c.f. Chovanec 1997) iff a  ðb  ða ^ bÞÞ exists (c.f. Riecˇanova´ 2000). In accordance with Riecˇanova´ (2000), we define a block in E as a maximal mutually compatible subset of E. Every element of E is in some block. By Chovanec (1997), an MV-algebra (introduced by Chang 1959) can be defined as a lattice effect algebra of mutually compatible elements. Riecˇanova´ (2000) proved that in a lattice effect algebra E, every block M of E has the following properties. 1. 2. 3. 4.

a; b 2 M implies that a ^ b; a _ b 2 M: a; b 2 M; a ? b imply that a  b 2 M: a 2 M implies that a0 2 M: M is an MV-algebra.

We note that (2) and (3) mean that every block M is a sub-effect algebra of E. In particular, a; b 2 M with a C b implies that a  b 2 M:

2 Ideals generated by ideals of the compatibility center Let E be a lattice effect algebra. The set KðEÞ ¼ fa : a $ b for all b 2 Eg is called the compatibility center of E. Since the compatibility center of a lattice effect algebra E is (obviously) the intersection of all blocks of E, it is easy to see that K(E) is a sub-effect algebra and, at the same time, a sublattice of E. Lemma 1

Given a; b 2 E; the following are equivalent.

(a) a $ b (b) a  ða0 ^ bÞ  b (c) a  ða0 ^ bÞ ¼ b  ðb ^ a0 Þ (d) a  ða ^ b0 Þ  b (e) a  ða ^ b0 Þ ¼ b  ðb ^ a0 Þ Proof Most of this Lemma was proved before by Chovanec (1997), Bennett (1995) and Riecˇanova´ (2000). The remaining bits follow by a simple application of some well-known ð; Þ ‘‘dualities’’ and are therefore omitted. h Lemma 2 Let a; b; a1 ; b1 2 E be such that a  a1 ; b  b1 ; b1 $ a; a1 ? b1 : Then a1  b1  a  ðb ^ a0 Þ: Proof Since a1 ? b1 ; a1 ¼ a1 ^ b1 0 : Thus, a1  b1 ¼ ða1 ^ b1 0 Þ  b1 : Since a1  a; ða1 ^ b1 0 Þ  b1  ða ^ b1 0 Þ  b1 : By Lemma 1, a $ b1 implies that ða ^ b1 0 Þ b1 ¼ a  ða0 ^ b1 Þ  a  ða0 ^ bÞ: h Lemma 3 Let a; b; z 2 E be such that a ? b; a $ z; b $ z: Then, ða  bÞ ^ z  ða ^ zÞ  ðb ^ zÞ:

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Proof It is implicit in Riecˇanova´ (2000), proof of Theorem 2.1, part (ii) that x $ z; y $ z; y  x imply that ðy ^ zÞ  ðx ^ zÞ  y  x: Obviously, ðy ^ zÞ  ðx ^ zÞ  z: Thus, under the above assumptions on x, y, z, we see that ðy ^ zÞ  ðx ^ zÞ  ðy  xÞ ^ z: By Riecˇanova´ (2000), a $ z and b $ z imply that a  b $ z: Thus, for y ¼ a  b and x = b, we have x; y $ z and y C x. The rest of the proof follows by a simple computation: ða ^ zÞ  ðb ^ zÞ ¼ ððða  bÞ  bÞ ^ zÞ  ðb ^ zÞ ¼ ððy  xÞ ^ zÞ  ðx ^ zÞ  ððy ^ zÞ  ðx ^ zÞÞ  ðx ^ zÞ ¼ ðy ^ zÞ ¼ ða  bÞ ^ z. h In what follows, we write hXiE for the smallest ideal of a lattice effect algebra E containing a nonempty set X; since the set of all ideals of an effect algebra forms a complete lattice with respect to inclusion, hXiE always exists. Clearly, if E is an MV-algebra, then hXi ¼ fy 2 E : y  x1      xn for some x1 ; . . .; xn 2 Xg: For a singleton, we write shortly hfugi ¼ hui: It is easy to see that in every MV-algebra we have hui \ hvi ¼ hu ^ vi: Proposition 1 Let G be a subalgebra of K(E), let I 2 IðGÞ: Then, hIiE equals fa1 2 E : a1  a for some a 2 Ig. Moreover, hIiE is a Riesz ideal in E. Proof The inclusion hIiE (1) is trivial. For the opposite inclusion, it suffices to prove that (1) is an ideal in E. It is obvious that (1) is an order ideal. Let a1 ; b1 2 (1); a1 ? b1 : There are a; b 2 I such that a1 B a and b1 B b. Since a; b 2 KðEÞ; the conditions of Lemma 2 are satisfied. Hence, a1  b1  a  ðb ^ a0 Þ 2 I: Therefore, a1  b1 2 ð1Þ: Let i; a; b 2 E be such that i 2 hIiE ; a ? b; i  a  b: By the above part of the proof, there is z 2 I such that i B z. Since i  a  b; i ^ z ¼ i  ða  bÞ ^ z: Since z 2 KðEÞ; we may apply Lemma 3 to obtain ða  bÞ ^ z  ða ^ zÞ  ðb ^ zÞ: Since a ^ z; b ^ z 2 hIiE ; hIiE is a Riesz ideal. h Proposition 2 Let G be a subalgebra of KðEÞ; u; v 2 E; u 6¼ v: There exists a prime ideal I in G such that u 6  hIi v: Proof Let {Ja} be a nonempty chain of ideals in G, each S of them satisfying u 6  hJa iE v: Let J ¼ fJa g: We claim that u 6  hJiE v: Indeed, assuming the contrary, there are j0 ; j1  j 2 J such that u  j0 ¼ v  j1 : Since j 2 J; j 2 Ja for some a, so u  hJa iE v which is a contradiction. Therefore, we may apply Zorn’s lemma to prove that there is

Congruences generated by ideals of the compatibility center of lattice effect algebras

I 2 IðGÞ such that I is maximal with respect to the property u 6  hIiE v: We shall prove that I 2 PIðGÞ: Assume the contrary, i.e., there are b; c 2 G such that b ^ c 2 I and b; c 62 I: Denote Ib ¼ I _ hbiG ; Ic ¼ I _ hciG ; where the joins are taken in I(G). By maximality of I, we have u  hIb i v and u  hIc i v: By Jencˇa (2000), Lemma 3, this implies that u  hIc i\hIb i v: Thus, there are i; j 2 hIc i \ hIb i such that u  i ¼ v  j: By Proposition 1, there are ic 2 Ic and ib 2 Ib such that i B ib, ic. Since i  ib ^ ic 2 Ib \ Ic ; i 2 hIb \ Ic i: Similarly, j 2 hIb \ Ic i; therefore u  hIb \Ic i v: Since I(G) is a distributive lattice, Ib \ Ic ¼ ðI _ hbiG Þ \ ðI _ hciG Þ ¼ I _ ðhbiG \ hciG Þ: Since G is an MV-algebra, hbiG \ hciG ¼ hb ^ ciG : By assumption, b ^ c 2 I: This implies that I _ hb ^ ciG ¼ I so u  hIi v which gives us the desired contradiction. h Owing to Proposition 2, the map Y E=hIi, b : E7! I2PIðKðEÞÞ

given by abðIÞ ¼ ½a I ; is injective. If E is an MV-algebra, then each E=hIi is totally ordered. As a consequence, we obtain Chang’s representation theorem for MV-algebras (Chang 1959) (which says that every MV-algebra is a subdirect product of totally ordered MV-algebras). On the other hand, Proposition 2 generalizes an OML-theoretic result by Graves and Selesnick from their paper (Graves 1973).

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Research and Development Agency under the contracts APVV-007310, APVV-0178-11.

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Acknowledgments This research is supported by grants VEGA G-2/0059/12,G-1/0297/11 of MSˇ SR, Slovakia, and by the Slovak

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