Page 131 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 131
IV: Skew Boolean Algebras
Theorem 4.2.4. Given a skew Boolean algebra S = ∑1r Di0 with finitely many D-classes
and with the Di being its atomic D-classes, then its congruence lattice decomposes as follows:
Con(S) ≅ Con( D10 ) × … × Con( Dr0 ). £
Recall that for a primitive algebra D0, Con( Dr0 ) is essentially the lattice of rectangular partitions
of D, augmented by a bottom element corresponding to the universal congruence on D0.
Free algebras: the finite case
Given a non-empty set X:
SBAX is the free skew Boolean algebra on X.
RSBAX is the free right-handed skew Boolean algebra on X.
LSBAX is the free left-handed skew Boolean algebra on X.
GBAX is the free generalized Boolean algebra on X.
Free algebras are, of course, unique to within isomorphism. Thus if we say “the free” we have in
mind a particular concrete instance, from which we are free (in an alternative sense) to find other
isomorphic variants. In this paper, the default free algebra FX on an alphabet X is the algebra of
all terms (or polynomials) in X. In the current context, the terms are defined inductively as
follows.
1) Each x in X is a term, as is the constant 0.
2) If u and v are terms, so are (u ∨ v), (u ∧ v) and (u \ v).
Two terms, u and v, are equivalent in FX iff u = v is an identity in the given variety of algebras.
Clearly these criteria for equivalence differ among the four varieties of interest. Given an SBA
equation of terms in X, u = v, one can check if it is a left-handed identity (or right-handed
identity) by seeing if it holds for all evaluations on 3L (or on 3R). It is an SBA identity precisely
when it holds for all evaluations on both 3L and 3R. Finally, it is a GBA identity if and only if it
holds for all evaluations on 2. In our considerations, we are free to relax aspects of the syntax for
parentheses if all ways of reinserting them lead to equivalent expressions. E.g., that would
happen with x∨y∨z, but not with x∧y∨z.
Given the universal character of the homomorphisms involved in the Clifford-McLean
and the Kimura Factorization theorems for skew Boolean algebras:
GBAX ≅ SBAX /D ≅ RSBAX /D ≅ LSBAX /D.
RSBAX ≅ SBAX /L and LSBAX ≅ SBAX /R.
SBAX ≅ LSBAX ×GBAX RSBAX.
(Indeed, let V be any variety of algebras with W a subvariety of V. For each algebra A in V, let
θA be the congruence on A such that A/θA is in W and the induced map ϕA : A → A/θA is a
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Theorem 4.2.4. Given a skew Boolean algebra S = ∑1r Di0 with finitely many D-classes
and with the Di being its atomic D-classes, then its congruence lattice decomposes as follows:
Con(S) ≅ Con( D10 ) × … × Con( Dr0 ). £
Recall that for a primitive algebra D0, Con( Dr0 ) is essentially the lattice of rectangular partitions
of D, augmented by a bottom element corresponding to the universal congruence on D0.
Free algebras: the finite case
Given a non-empty set X:
SBAX is the free skew Boolean algebra on X.
RSBAX is the free right-handed skew Boolean algebra on X.
LSBAX is the free left-handed skew Boolean algebra on X.
GBAX is the free generalized Boolean algebra on X.
Free algebras are, of course, unique to within isomorphism. Thus if we say “the free” we have in
mind a particular concrete instance, from which we are free (in an alternative sense) to find other
isomorphic variants. In this paper, the default free algebra FX on an alphabet X is the algebra of
all terms (or polynomials) in X. In the current context, the terms are defined inductively as
follows.
1) Each x in X is a term, as is the constant 0.
2) If u and v are terms, so are (u ∨ v), (u ∧ v) and (u \ v).
Two terms, u and v, are equivalent in FX iff u = v is an identity in the given variety of algebras.
Clearly these criteria for equivalence differ among the four varieties of interest. Given an SBA
equation of terms in X, u = v, one can check if it is a left-handed identity (or right-handed
identity) by seeing if it holds for all evaluations on 3L (or on 3R). It is an SBA identity precisely
when it holds for all evaluations on both 3L and 3R. Finally, it is a GBA identity if and only if it
holds for all evaluations on 2. In our considerations, we are free to relax aspects of the syntax for
parentheses if all ways of reinserting them lead to equivalent expressions. E.g., that would
happen with x∨y∨z, but not with x∧y∨z.
Given the universal character of the homomorphisms involved in the Clifford-McLean
and the Kimura Factorization theorems for skew Boolean algebras:
GBAX ≅ SBAX /D ≅ RSBAX /D ≅ LSBAX /D.
RSBAX ≅ SBAX /L and LSBAX ≅ SBAX /R.
SBAX ≅ LSBAX ×GBAX RSBAX.
(Indeed, let V be any variety of algebras with W a subvariety of V. For each algebra A in V, let
θA be the congruence on A such that A/θA is in W and the induced map ϕA : A → A/θA is a
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