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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.

##### MSC:
 06D35 MV-algebras 08B15 Lattices of varieties 08A30 Subalgebras, congruence relations
##### Keywords:
varieties; order; ordinal sum; lattice of subvarieties; BL-algebras
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