A New Representation of Finite Hoops Using a New Type of Product of Structures

Abstract We define a new type of product of hoops which, in the case of finite hoops, can decompose an arbitrary hoop $$\textbf{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> into a filter F and the corresponding homomorphic image $$\textbf{A}/F$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> . Since our new product is associative up to isomorphism, we can show that every finite hoop is representable as a product of finite MV-chains.