Some varieties containing relation algebras

Abstract

Three varieties of algebras are introduced which extend the variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper A"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">RA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of relation algebras. They are obtained from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper A"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">RA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N upper A"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">NA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W upper A"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">WA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper A"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">SA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W upper A"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">WA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W upper A"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">WA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the closure of the variety of representable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper A"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">RA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ’s under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples.