Page 89 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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III: QUASILATTICES, PARALATTICES & THEIR
CONGRUENCES
We take a closer look at quasilattices, paralattices, and especially refined quasilattices
which by definition are simultaneously algebras of both types. Particular attention is given to
their congruence lattices and to related topics such as Green’s equivalences and simple algebras.
Since all skew lattices are refined quasilattices, our study has implications for skew lattices.
In the first section we consider quasilattices where D(∨) = D(∧), with D denoting the
common congruence. These are the noncommutative lattices for which the Clifford-McLean
Theorem holds: the maximal lattice image of a quasilattice Q is Q/D and its maximal antilattice
subalgebras are its D-classes. The congruence D also plays an major role in the congruence
lattice Con(Q) of a quasilattice Q. For instance, given a family of congruences {θi}, both
D ∧ supi(θi) = supi(D ∧ θi)
and its dual hold. (Theorem 3.1.1) As a consequence, Con(Q) is a subdirect product of the
interval sublattices [D, ∇] and [Δ,D], where ∇ is the universal congruence where Δ is the identity
congruence. (Theorem 3.1.2) In particular, Con(Q) is naturally a copy of the direct product [D,
∇] × [Δ,D] precisely when Q itself factors as the direct product T × A of a lattice T and an
antilattice A. (Theorem 3.1.3) Due to these theorems, the only simple quasilattices (in that the
congruence lattice reduces to {Δ, ∇}) are simple lattices and simple antilattices.
Section 2 addresses the topic of simple antilattices. Its main result Theorem 3.2.3 states
that finite simple antilattices exist for all composite orders greater than 5. Antilattices of odd
prime order are trivially non-simple. In the remaining cases, algebras of orders 1 or 2 are always
simple, while simple antilattices of order 4 are shown to be impossible.
In the next section we look at noncommutative lattices that are regular in the strongest
sense: L(∧), R(∧), L(∨) and R(∨) are congruences relative to both ∨ and ∧. Flat quasilattices
(where D(∨) is either L(∨) or R(∨) and likewise D(∧) is either L(∧) or R(∧), with the two remaining
Green’s relations being equality) are trivially regular since D is both unambiguous and a
congruence on quasilattices (Theorem 3.3.2). Moreover, every regular quasilattice factors as the
fibered product of its four possible maximal flat images (Theorem 3.2.4).
In Section 4 we study paralattices, and especially refined quasilattices that are paralattices
and quasilattices simultaneously. A paralattice S is also a quasilattice if and only if both D(∨) and
D(∧) are congruences, in which case they meld into a single relation D. A regular paralattice is
87
CONGRUENCES
We take a closer look at quasilattices, paralattices, and especially refined quasilattices
which by definition are simultaneously algebras of both types. Particular attention is given to
their congruence lattices and to related topics such as Green’s equivalences and simple algebras.
Since all skew lattices are refined quasilattices, our study has implications for skew lattices.
In the first section we consider quasilattices where D(∨) = D(∧), with D denoting the
common congruence. These are the noncommutative lattices for which the Clifford-McLean
Theorem holds: the maximal lattice image of a quasilattice Q is Q/D and its maximal antilattice
subalgebras are its D-classes. The congruence D also plays an major role in the congruence
lattice Con(Q) of a quasilattice Q. For instance, given a family of congruences {θi}, both
D ∧ supi(θi) = supi(D ∧ θi)
and its dual hold. (Theorem 3.1.1) As a consequence, Con(Q) is a subdirect product of the
interval sublattices [D, ∇] and [Δ,D], where ∇ is the universal congruence where Δ is the identity
congruence. (Theorem 3.1.2) In particular, Con(Q) is naturally a copy of the direct product [D,
∇] × [Δ,D] precisely when Q itself factors as the direct product T × A of a lattice T and an
antilattice A. (Theorem 3.1.3) Due to these theorems, the only simple quasilattices (in that the
congruence lattice reduces to {Δ, ∇}) are simple lattices and simple antilattices.
Section 2 addresses the topic of simple antilattices. Its main result Theorem 3.2.3 states
that finite simple antilattices exist for all composite orders greater than 5. Antilattices of odd
prime order are trivially non-simple. In the remaining cases, algebras of orders 1 or 2 are always
simple, while simple antilattices of order 4 are shown to be impossible.
In the next section we look at noncommutative lattices that are regular in the strongest
sense: L(∧), R(∧), L(∨) and R(∨) are congruences relative to both ∨ and ∧. Flat quasilattices
(where D(∨) is either L(∨) or R(∨) and likewise D(∧) is either L(∧) or R(∧), with the two remaining
Green’s relations being equality) are trivially regular since D is both unambiguous and a
congruence on quasilattices (Theorem 3.3.2). Moreover, every regular quasilattice factors as the
fibered product of its four possible maximal flat images (Theorem 3.2.4).
In Section 4 we study paralattices, and especially refined quasilattices that are paralattices
and quasilattices simultaneously. A paralattice S is also a quasilattice if and only if both D(∨) and
D(∧) are congruences, in which case they meld into a single relation D. A regular paralattice is
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