Page 66 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Given a primitive skew lattice P, each component of P has nonempty intersection with
each coset of P. Thus we say that a primitive skew lattice P is maximally disconnected if distinct
elements of each coset of P belong to distinct components of P. P is degenerate if it is both
connected and maximally disconnected. If P has D-class structure A > B, then P is degenerate
precisely when a > b for all a ∈A and all b ∈B. Finally, a coset component of P is any
(necessarily nonempty) intersection of a coset in P with a component of P. For primitive skew
lattices we have the following results:
Lemma 2.4.4. Given a primitive skew lattice P, the cosets of P form a congruence
partition of P. If ζ is the corresponding congruence, then the primitive skew lattice P/ζ is the
maximal degenerate image of P.
Proof. Given our descriptions of primitive skew lattices in Theorems 2.4.1 and 2.4.2 above,
pointwise computations yield Ai ∨ Bj = Ai = Bj ∨ Ai and Ai ∧ Bj = Bj = Bj ∧ Ai. £
Theorem 2.4.5. If P is a primitive skew lattice, then its components are the maximal
connected subalgebras of P. Moreover, they are the congruence classes of the congruence ρ
whose quotient algebra P/ρ is the maximal rectangular image of P. Thus the coset components of
P form the congruence partition for the congruence ζ∩ρ for which the quotient algebra P/ζ∩ρ is
the maximal disconnected image of P. Finally, if P is maximally disconnected, then it factors as
the product of a degenerate skew lattice with a rectangular skew lattice.
Proof. The components of P clearly induce an equivalence relation ρ on P. To begin, let σ
denote the symmetric closure of ≥. Thus a σ b means that either a ≥ b or b ≥ a. Since ρ is the
transitive closure of σ, to show that ρ is a congruence we need only show that a σ b and aʹ σ bʹ
imply that both a∨aʹ ρ b∨bʹ and a∧aʹ ρ b∧bʹ. We first show this under the added restriction that
P is right primitive. Thus a∨aʹ ≥ a and b∨bʹ ≥ b so that a∨aʹ ρ b∨bʹ follows. Similarly, a∧aʹ ≤ aʹ
and b∧bʹ ≤ bʹ so that a∧aʹ ρ b∧bʹ also follows and ρ is shown to be a congruence. Similarly, if P
is left primitive then again, ρ is a congruence. By the Kimura factorization, ρ is a congruence on
any primitive skew lattice. The first part of the theorem follows. Since the coset components
arise as congruence classes for the meet congruence ζ∩ρ, the second part of the theorem is seen.
Finally the assumption of being maximally disconnected insures that the induced homomorphism
from P to P/ζ × P/ρ with kernel congruence ζ∩ρ is an isomorphism. £
A skew lattice S is bounded if it has a maximal class A and a minimal class Z in which
case the primitive skew lattice A∪Z forms the boundary Bd(S) of S. A generalization of first part
of the above theorem complements the Clifford-McLean Theorem.
Theorem 2.4.6. The components of a skew lattice S are its maximal connected sub-
algebras. Moreover, their partition of S is the congruence class partition into for the congruence
ρ for which the induced quotient algebra S/ρ is the maximal rectangular image of S. If S is also
bounded with boundary algebra Bd(S), then the inclusion Bd(S) ⊆ S induces an isomorphism of
maximal rectangular images.
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Given a primitive skew lattice P, each component of P has nonempty intersection with
each coset of P. Thus we say that a primitive skew lattice P is maximally disconnected if distinct
elements of each coset of P belong to distinct components of P. P is degenerate if it is both
connected and maximally disconnected. If P has D-class structure A > B, then P is degenerate
precisely when a > b for all a ∈A and all b ∈B. Finally, a coset component of P is any
(necessarily nonempty) intersection of a coset in P with a component of P. For primitive skew
lattices we have the following results:
Lemma 2.4.4. Given a primitive skew lattice P, the cosets of P form a congruence
partition of P. If ζ is the corresponding congruence, then the primitive skew lattice P/ζ is the
maximal degenerate image of P.
Proof. Given our descriptions of primitive skew lattices in Theorems 2.4.1 and 2.4.2 above,
pointwise computations yield Ai ∨ Bj = Ai = Bj ∨ Ai and Ai ∧ Bj = Bj = Bj ∧ Ai. £
Theorem 2.4.5. If P is a primitive skew lattice, then its components are the maximal
connected subalgebras of P. Moreover, they are the congruence classes of the congruence ρ
whose quotient algebra P/ρ is the maximal rectangular image of P. Thus the coset components of
P form the congruence partition for the congruence ζ∩ρ for which the quotient algebra P/ζ∩ρ is
the maximal disconnected image of P. Finally, if P is maximally disconnected, then it factors as
the product of a degenerate skew lattice with a rectangular skew lattice.
Proof. The components of P clearly induce an equivalence relation ρ on P. To begin, let σ
denote the symmetric closure of ≥. Thus a σ b means that either a ≥ b or b ≥ a. Since ρ is the
transitive closure of σ, to show that ρ is a congruence we need only show that a σ b and aʹ σ bʹ
imply that both a∨aʹ ρ b∨bʹ and a∧aʹ ρ b∧bʹ. We first show this under the added restriction that
P is right primitive. Thus a∨aʹ ≥ a and b∨bʹ ≥ b so that a∨aʹ ρ b∨bʹ follows. Similarly, a∧aʹ ≤ aʹ
and b∧bʹ ≤ bʹ so that a∧aʹ ρ b∧bʹ also follows and ρ is shown to be a congruence. Similarly, if P
is left primitive then again, ρ is a congruence. By the Kimura factorization, ρ is a congruence on
any primitive skew lattice. The first part of the theorem follows. Since the coset components
arise as congruence classes for the meet congruence ζ∩ρ, the second part of the theorem is seen.
Finally the assumption of being maximally disconnected insures that the induced homomorphism
from P to P/ζ × P/ρ with kernel congruence ζ∩ρ is an isomorphism. £
A skew lattice S is bounded if it has a maximal class A and a minimal class Z in which
case the primitive skew lattice A∪Z forms the boundary Bd(S) of S. A generalization of first part
of the above theorem complements the Clifford-McLean Theorem.
Theorem 2.4.6. The components of a skew lattice S are its maximal connected sub-
algebras. Moreover, their partition of S is the congruence class partition into for the congruence
ρ for which the induced quotient algebra S/ρ is the maximal rectangular image of S. If S is also
bounded with boundary algebra Bd(S), then the inclusion Bd(S) ⊆ S induces an isomorphism of
maximal rectangular images.
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