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Boolean semilatticesBoolean semilattices|
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complex algebra of S is Cm(S) = (P(S),∪,Ø, ∩,S,−,·), where \langle P(S),\cup,\emptyset, |
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complex algebra of S is Cm(S) = (P(S),∪,Ø,∩,S,−,·), where $\langle P(S),\cup,\emptyset, |
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X·Y = {x·y | x ∈ X, y ∈ Y}. |
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X·Y = {x·y | x ∈ X, y ∈ Y}. |
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[Size 8]?: ge 96 out of 104 |
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[Size 8]?: ≥ 97 out of 104 |
A Boolean semilattice is a structure A = (A,∨,0, ∧,1,¬,·) such that A is in the variety generated by complex algebras of semilattices.
Let S = (S,·) be a semilattice. The complex algebra of S is Cm(S) = (P(S),∪,Ø,∩,S,−,·), where (P(S),∪,Ø, ∩,S,−) is the Boolean algebra of subsets of S, and
X·Y = {x·y | x ∈ X, y ∈ Y}.
Remark: The complex algebra of any groupoid is a Boolean algebra with one binary operator.
Let A and B be Boolean semilattices. A morphism from A to B is a function h : A→B that is a Boolean homomorphism and preserves ·: h(x·y) = h(x)·h(y).
| Classtype | variety |
| [Finitely axiomatizable]? | open |
| Equational theory | |
| Quasiequational theory | |
| First-order theory | |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | yes, n = 2 |
| Congruence regular | yes |
| Congruence uniform | |
| Congruence extension property | yes |
| Definable principal congruences | |
| Equationally definable principal congruences | |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
[Some members of BSlat]?