# zbMATH — the first resource for mathematics

Cyclic ordered groups and MV-algebras. (English) Zbl 0795.06015
A cyclically ordered group is a system $$(G,+,-,0,T)$$, where $$(G,+,-,0)$$ is a group and $$T$$ is a ternary relation which fulfills for all $$a$$, $$b$$, $$c$$, $$d\in G$$: C1. If $$a\neq b\neq c\neq a$$ then exactly one of $$T(a,b,c)$$ and $$T(a,c,b)$$ holds; C2. $$T(a,b,c)\Rightarrow a\neq b\neq c\neq a$$; C3. $$T(a,b,c)\Rightarrow T(c,a,b)$$; C4. $$T(b,c,a) \& T(c,d,a)\Rightarrow T(b,d,a)$$; C5. $$T(a,b,c)\Rightarrow T(d+ a,d+ b,d+c) \& T(a+ d,b+ d,c+ d)$$. A partially cyclically ordered group is a system $$(G,+,-,0,T)$$, where the axioms C3, C4, C5 and C1p. $$T(a,b,c)\Rightarrow\neg T(a,c,b)$$; C6. $$T(a,b,c)\Rightarrow T(-c, -b, -a)$$ hold. An MV-algebra is a system $$(A,\oplus,*,\neg,0,1)$$ verifying: the addition is associative, commutative and 0 is its zero, $$x\oplus 1=1$$, $$\neg\neg x=x$$, $$\neg 0=1$$, $$x\oplus\neg x=1$$, $$\neg(\neg x\oplus y)\oplus y= \neg(x\oplus \neg y)\oplus x$$, $$x* y=\neg(\neg x\oplus\neg y)$$. It is proved that any MV- algebra $$A$$ can be obtained from an abelian $$\ell$$-group with strong unit $$u\in G= (G,\lor,\land,+, -,0, u)$$ by defining: $$A= [0,u]= \{a\mid 0\leq a\leq u\}$$; $$a\oplus b= (a+ b)\land u$$; $$\neg a= u-a$$ and $$1= u$$. For any pco-group $$G$$, a partial order can be defined by $$(*)$$ $$a\leq b$$ iff $$a= b$$ or $$T(0,a,b)$$ or $$a=0$$. A pco-group $$G$$ will be called a lattice- cyclical group, if, for the order defined in $$(*)$$ the structure $$(G,0,\leq)$$ admits a distributive lattice structure with first element. An lc-group $$G$$ is called projectable if one can define a binary operation pr on $$G$$, compatible for the left argument with the group operations, such that $$h'= \text{pr}(g,h)$$ implies $$h'\in h^ \perp$$ and $$g- h'\in h^{\perp\perp}$$, where $$h^ \perp= \{a\mid a\land h=0\}$$. The main results of the paper. Let $$G$$ be a projectable lc-group with weak unit. There exists an $$\ell$$-group $$G'$$ with a strong unit $$u$$ such that $$G\simeq G'$$. Call LC and MV, the categories of projectable lc- groups with weak unit and projectable MV-algebras, respectively. Theorem. The categories LC and MV are equivalent.
Reviewer: F.Šik (Brno)

##### MSC:
 06F15 Ordered groups 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
Full Text:
##### References:
 [1] R. Balbes and P. Dwinger: Distributive lattices. University of Missouri Press, 1974. · Zbl 0321.06012 [2] A. Bigard, K. Keimel and S. Wolfenstein: Groupes et anneaux réticulés. Springer LNM 608 (1978). · Zbl 0384.06022 [3] S. Burris and H. P. Sankappanavar: A course in universal algebra. Springer-Verlag, New York-Heidelberg-Berlin, 1981. · Zbl 0478.08001 [4] C. C. Chang: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88 (1958), 467-490. · Zbl 0084.00704 [5] C. C. Chang: A new proof of the completeness of the Łukasiewicz axioms. Trans. Am. Math. Soc. 93 (1959), 74-80. · Zbl 0093.01104 [6] J. Font, A. Rodríguez and A. Torrens: Wajsberg algebras. Stochastica, VIII (1984), 5-31. · Zbl 0557.03040 [7] F. Lacava: Alcune proprietà delle ł-algebre e delle t-algebre essistenzialmente chiuse. Boll. Unione Mat. Italiana (5) 16A (1979), 360-366. · Zbl 0427.03024 [8] D. Mundici: Interpretation of $$C^*$$-Algebras in Łukasiewicz sentential calculus. J. Func. An. 65 (1986), 15-63. · Zbl 0597.46059 [9] L. Rieger: On ordered and cyclically ordered groups I, II, III. Věstník král. české spol. nauk (1946 1947 1948), 1-31 1-33 1-26, In Czech. [10] S. Swierczkowski: On cyclically ordered groups. Fund. Math. 47 (1959), 161-166. · Zbl 0096.01501 [11] A. Torrens: W-algebras which are Boolean products of members of SR[1] and CW-algebras. Studia Logica, XXLVI 33 (1987), 263-272. · Zbl 0621.03042 [12] V. Weispfenning: Elimination of quantifiers for certain ordered and lattice-ordered abelian groups. Bull. Soc. Math. Belg., XXXIII (1981), Fasc. I, serie B, 131-155. · Zbl 0499.03012 [13] A. I. Zabarina and G. G. Pestov: Swierczkowski’s theorem. Sibirski Math. J. 25 (1984), 46-53. · Zbl 0579.06014 [14] S. D. Zheleva: Cyclically ordered groups. Sibirski Math. J. 17 (1976), 1046-1051. · Zbl 0362.06022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.