Varieties of BL-algebras. I: General properties. (English) Zbl 1034.06009

The aim of the paper is to show some facts and techniques that are useful in order to describe the lattice of subvarieties of BL-algebras. An algebra \((A,\to,\cdot,0,1)\) is called a BL-algebra if \((A,\cdot,1)\) is a commutative monoid and the following identities are satisfied: \(x\to x= 1\), \(x\cdot(x\to y)= y\cdot(y\to x)\), \(x\to (y\to z)= (x\cdot y)\to z\), \(0\to x=1\) and \(((x\to y)\to z)\to (((y\to x)\to z)\to z)= 1\). Some special cases of BL-algebras are introduced; for example a BL-algebra satisfying the equation \(x\cdot x= x\) is called a Gödel algebra. The results of the paper include a description of subalgebras and homomorphic images of totally ordered BL-algebras and a characterization of totally ordered BL-algebras that generate the variety of all BL-algebras and other results.


06D35 MV-algebras
08B15 Lattices of varieties
08A30 Subalgebras, congruence relations
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