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The Variety of Commutative BCI-Algebras is 2-Based
Authors:Tingrong Zou
Institution:(1) Department of Mathematics, Mianyang Teacherrsquos College, Sichuan, Mianyang, 621000, China
Abstract:In this note, we first solve the following open problem in 5]: Can the variety of commutative BCI-algebras be defined by two identities? An algebra of type (2, 0) is a commutative BCI-algebra if and only if it satisfies $u\ast\left(((x \ast y) \ast (x \ast y))(z \ast y)\right) = uIn this note, we first solve the following open problem in 5]: Can the variety of commutative BCI-algebras be defined by two identities? An algebra of type (2, 0) is a commutative BCI-algebra if and only if it satisfies $$u\ast\left(((x \ast y) \ast (x \ast y))(z \ast y)\right) = u$$ and $$x \ast (x \ast y)= \left(y \ast (y \ast (x \ast (x \ast y)))\right) \ast 0.$$ (see Theorem 2 below).Next, we prove that I-variety 2] is also 2-based. Finally, we show that I-variety is a proper subvariety of the variety of commutative BCI-algebras.AMS Subject Classification (2000): 03G25, 06A10, 06D99
Keywords:commutative BCI-algebra  algebra variety  equational base
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