The follwing theorem proven by O.~Frink in Pseudo-complements in semi-lattices, Duke Math. J. 1962, Vol. 29, may help:

If $\langle S;\wedge,-\rangle$ is a pseudo-complemented meet-semilattice, then the subset $B(S):=${$-x:x\in S$} with the order inherited from $S$ is a boolean algebra.( A proof of this result can also be found in Grätzer's General Lattice Theory.)

As $-$ is a pseudo-complement on $S$ we have by definition $a\wedge b=0$ iff $b\leq -a$. This implies $a\leq b\rightarrow -a\geq -b$ and $---a=-a$. Therefore $a\mapsto -a$ is an order-reversing involution on $B(S)$.

In your case we have $B(M)=M$ since $--a=a$ for all $a\in M$. Therefore, $M$ is a boolean algebra.

The follwing theorem proven by O. Frink in Pseudo-complements in semi-lattices, Duke Math. J. 1962, Vol. 29, may help:

If $\langle S;\wedge,-\rangle$ is a pseudo-complemented meet-semilattice, then the subset $B(S):=${$-x:x\in S$} with the order inherited from $S$ is a boolean algebra.( A proof of this result can also be found in Grätzer's General Lattice Theory.)

As $-$ is a pseudo-complement on $S$ we have by definition $a\wedge b=0$ iff $b\leq -a$. This implies $a\leq b\rightarrow -a\geq -b$ and $---a=-a$. Therefore $a\mapsto -a$ is an order-reversing involution on $B(S)$.

In your case we have $B(M)=M$ since $--a=a$ for all $a\in M$. Therefore, $M$ is a boolean algebra.