Semi-lattice

2020 Mathematics Subject Classification: Primary: 06A12 [MSN][ZBL]

A commutative idempotent semi-group, that is, a semi-group satisfying the identities $x+y=y+x$ and $x+x=x$. Every semi-lattice $p=\langle p,+\rangle$ can be turned into a partially ordered set (the partial order $\leq$ is defined by the relation $a\leq b$ if and only if $a+b=b$) in which for any pair of elements there is a least upper bound $\sup\{a,b\}=a+b$. Conversely, every partially ordered set with least upper bounds for every pair of elements is a semi-lattice with respect to the operation $a+b=\sup\{a,b\}$. In this case one says that the partially ordered set is an upper semi-lattice (or a join semi-lattice, or a $\vee$-semi-lattice). A lower semi-lattice, also called a meet semi-lattice or a $\wedge$-semi-lattice, is dually defined as a partially ordered set in which any two elements have a greatest lower bound.

Comments

A band is a semi-group every element of which is idempotent (cf. also Band of semi-groups) (which is a decomposition of a semi-group into sub-semi-groups forming a band). Thus, an upper (lower) semi-lattice defines a commutative band, and conversely.

References

[a1]	A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , 1 , Amer. Math. Soc. (1961) pp. §1.8 Zbl 0111.03403

Comments

The free join semilattice on a set $X$ is the set of all finite subsets of $X$ ordered by inclusion.

References

[b1]	Peter T. Johnstone Sketches of an elephant Oxford University Press (2002) ISBN 0198534256 Zbl 1071.18001