nLab 01-bounded semilattice

This article is about bounded semilattices in the sense of the semilattice having both a bottom element and a top element. For “bounded semilattices” in the sense of a join-semilattice having a bottom element or a meet-semilattice having a top element, see semilattice.

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Definition

A 01-bounded semilattice $L$ is a semilattice which is also a bounded poset with respect to the semilattice-induced partial order, with bottom element $0 \in L$ and top element $1 \in L$.

Equivalently, $L$ is a semilattice with an absorbing element and a neutral element with respect to the semilattice operation. The absorption and neutral axioms automatically imply that the semilattice is bounded. $L$ is called a 01-bounded join-semilattice if $0$ is the neutral element and $1$ is the absorbing element, and $L$ is called a 01-bounded meet-semilattice if $1$ is the neutral element and $0$ is the absorbing element

The Lawvere theory of 01-bounded semilattices is the semilattice cube category, the cartesian cube category with one connection.

Examples

• Every lattice is both a 01-bounded meet-semilattice and a 01-bounded join-semilattice

• Every Boolean rig as defined by Fernando Guzmán is a 01-bounded meet-semilattice.

• Every commutative multiplicatively idempotent rig is a 01-bounded meet-semilattice

• Every additively idempotent rig is a 01-bounded join-semilattice.

Related concepts

• lattice

• semilattice

• Boolean rig

• Boolean ring

• category of cubes

• Phoa's principle

References

• Evan Cavallo and Christian Sattler, Relative elegance and cartesian cubes with one connection (2022). [arXiv:2211.14801]

• Leoni Pugh, Jonathan Sterling, When is the partial map classifier a Sierpiński cone? (arXiv:2504.06789)