  A two-dimensional automaton operates on arrays of symbols. While a standard
(four-way) two-dimensional automaton can move its input head in four
directions, restricted two-dimensional automata are only permitted to move
their input heads in three or two directions; these models are called three-way
and two-way two-dimensional automata, respectively.
  In two dimensions, we may extend the notion of concatenation in multiple
ways, depending on the words to be concatenated. We may row-concatenate (resp.,
column-concatenate) a pair of two-dimensional words when they have the same
number of columns (resp., rows). In addition, the diagonal concatenation
operation combines two words at their lower-right and upper-left corners, and
is not dimension-dependent.
  In this paper, we investigate closure properties of restricted models of
two-dimensional automata under various concatenation operations. We give
non-closure results for two-way two-dimensional automata under row and column
concatenation in both the deterministic and nondeterministic cases. We further
give positive closure results for the same concatenation operations on unary
nondeterministic two-way two-dimensional automata. Finally, we study closure
properties of diagonal concatenation on both two- and three-way two-dimensional
automata.
