Varieties of n-groups
Wieslaw A. Dudek · 2024
A group can be defined as a semigroup in which the equations https://www.w3.org/1998/Math/MathML" display="inline"> a x = b https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781032703541/24134cf2-9c89-4d99-9d1c-133181f5309f/content/math2_1.tif "/> and https://www.w3.org/1998/Math/MathML" display="inline"> y a = b https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781032703541/24134cf2-9c89-4d99-9d1c-133181f5309f/content/math2_2.tif "/> have solutions, but it can also be defined as an algebra satisfying several (or even one) identities. The class of groups is therefore a variety. This means that this class is closed under the taking of homomorphic images, subalgebras, and direct products.