TITLE: Do cardinals in ZF have lattice operations?
QUESTION [5 upvotes]: The answer to the question in the title is probably "no", although the cardinal sum appears to have many of the properties of the lattice join. But it would be nice to have an explicit example of failure of the lattice meet and join properties.

Does $\sf ZF$ prove the existence of greatest lower bounds and/or least upper bounds for pairs of (not necessarily well-ordered) cardinals $\frak m,n$?

With $\sf AC$, the answer is clearly "yes", because then the cardinals are totally ordered so for any pair of cardinals one is the meet and the other is the join.

REPLY [2 votes]: A reasonable follow-up question is whether there are some natural algebraic properties that the class of cardinals satisfies (provably in $\mathsf{ZF}$ or in $\mathsf{ZF}$ together with a weak axiom of choice). This is a natural problem and was investigated by Tarski in the 1940s, see 

MR0029954 (10,686f). Tarski, Alfred. Cardinal Algebras. With an Appendix: Cardinal Products of Isomorphism Types, by Bjarni Jónsson and Alfred Tarski. Oxford University Press, New York, N. Y., 1949. xii+326 pp. 

Tarski isolated such a set of properties under the notion of a cardinal algebra. A cardinal algebra consists of an abelian semigroup $(A,+)$ with identity together with an infinitary addition operation $\sum\!:A^{\mathbb N}\to A$, subject to some axioms. The theory found an unexpected recent application in descriptive set theory, in the study of Borel equivalence relations and their associated Borel cardinalities, see 

MR3549382. Kechris, Alexander S.; Macdonald, Henry L. Borel equivalence relations and cardinal algebras. Fund. Math. 235 (2016), no. 2, 183–198. 

In the paper by Kechris and Macdonald, the axioms are described in a somewhat different (but equivalent) way than in Tarski's book, namely: 

$\sum_{n=0}^\infty \mathfrak{m}_n=\mathfrak m_0+\sum_{n=0}^\infty \mathfrak m_{n+1}$.
$\sum_n (\mathfrak m_n+\mathfrak n_n)=\sum_n \mathfrak m_n+\sum_n\mathfrak n_n$.
If $\mathfrak m+\mathfrak n=\sum_n\mathfrak o_n$, then there are sequences $(\mathfrak m_n)_{n=0}^\infty$ and $(\mathfrak n_n)_{n=0}^\infty$ such that $\mathfrak m=\sum_n \mathfrak m_n$, $\mathfrak n=\sum_n \mathfrak n_n$, and $\mathfrak m_n+\mathfrak n_n=\mathfrak o_n$ for all $n$.
If two sequences $(\mathfrak m_n)$ and $(\mathfrak n_n)$ satisfy $\mathfrak m_n=\mathfrak n_n+\mathfrak m_{n+1}$ for all $n$, then there is an $\mathfrak o$ such that, for all $n$, $\mathfrak m_n=\mathfrak o+\sum_k \mathfrak n_{n+k}$. 

Tarski proves in his book that many properties of cardinal numbers provable in $\mathsf{ZF}$ or even $\mathsf{ZF}+\mathsf{AC}_\omega$ follow from his postulates (and some additional properties were later verified by Truss). He also provides several examples of cardinal algebras. 
What Kechris and Macdonald do is to note that there are natural operations $+$ and $\sum$ on the class of Borel equivalence relations on standard Borel spaces such that certain subclasses are cardinal algebras (for instance, the class of countable Borel equivalence relations, the class of treeable countable Borel equivalence relations, and others) . This results in proofs of new facts about Borel equivalence relations that had been open for years. For instance, if $E$ and $F$ are countable Borel equivalence relations and $nE\sim_B nF$, then $E\sim_B F$. Here, $\sim_B$ is the relation of Borel bireducibility. For another example, any sequence $F_0\le_B F_1\le_B \dots$ of countable Borel equivalence relations admits a least upper bound under Borel reducibility $\le_B$.