  The paper investigates the Lattice Agreement (LA) problem in asynchronous
systems. In LA each process proposes an element $e$ from a predetermined
lattice, and has to decide on an element $e'$ of the lattice such that $e \leq
e'$. Moreover, decisions of different processes have to be comparable (no two
processes can decide two elements $e$ and $e'$ such that $(e \not\leq e') \land
(e' \not\leq e)$). It has been shown that Generalized LA (i.e., a version of LA
proposing and deciding on sequences of values) can be used to build a
Replicated State Machine (RSM) with commutative update operations. The key
advantage of LA and Generalized LA is that they can be solved in asynchronous
systems prone to crash-failures (this is not the case with standard Consensus).
In this paper we assume Byzantine failures. We propose the Wait Till Safe (WTS)
algorithm for LA, and we show that its resilience to $f \leq (n-1)/3$
Byzantines is optimal. We then generalize WTS obtaining a Generalized LA
algorithm, namely GWTS. We use GWTS to build a RSM with commutative updates.
Our RSM works in asynchronous systems and tolerates $f \leq (n-1)/3$ malicious
entities. All our algorithms use the minimal assumption of authenticated
channels. When the more powerful public signatures are available, we discuss
how to improve the message complexity of our results (from quadratic to linear,
when $f={\cal O}(1)$). At the best of our knowledge this is the first paper
proposing a solution for Byzantine LA that works on any possible lattice, and
it is the first work proposing a Byzantine tolerant RSM built on it.
