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)


06F15 Ordered groups
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
Full Text: EuDML


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