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Characterization theorem of lattice implication algebras. (English) Zbl 1099.03054

By a lattice implication algebra we mean an algebra \((L;\wedge ,\vee ,\rightarrow ,^{\prime},0,1)\) of type \((2,2,2,1,0,0)\) such that: (1) \((L;\wedge ,\vee ,\rightarrow ,^{\prime },0,1)\) is a bounded lattice; (2) The unary operation \(^{\prime }\) is an order-reversing involution; (3) For all \(x,y,z\in L\) we have: \(x\rightarrow (y\rightarrow z)=y\rightarrow (x\rightarrow z);x\rightarrow x=1;x\rightarrow y=y^{\prime }\rightarrow x^{\prime };x\rightarrow y=y\rightarrow x=1\) implies \( x=y;(x\rightarrow y)\rightarrow y=(y\rightarrow x)\rightarrow x;(x\vee y)\rightarrow z=(x\rightarrow z)\wedge (y\rightarrow z);(x\wedge y)\rightarrow z=(x\rightarrow z)\vee (y\rightarrow z).\) In this paper the authors show that the class of all lattice implication algebras coincides with the class of all bounded commutative BCK-algebras and hence it is categorically equivalent to the class of MV-algebras and to the class of Wajsberg algebras.

MSC:

03G10 Logical aspects of lattices and related structures
06F35 BCK-algebras, BCI-algebras
06D35 MV-algebras
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