## 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)
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### References:

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