By Ladis D. Kovach

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Example text

There are many kinds of drawings for the infinite antichain, one way is to draw a circle on the plane and then to put the vertices of angular regions on the circle. This yields a description of an infinite 33 antichain. Similarly, by using translations on the plane, we can draw an infinite chain. ------ anti chain chain It has proved that every interval order is an angle order ([FiTr]). ". It is interesting to draw angle orders on the plane for important posets, and to find a poset which can not be represented as an angle order.

There are three 51 antichains, viz, {1, 2}, { 1, 4} and {3, 4} in the letter N poset. The number of ways of sending the elements of X is 24 - 2 = 14, and hence there are 3 x 14 =1 onto the doubleton = 42 Harris maps of this type. Thus there are a total of 4 + 42 = 46 Harris mappings from X =1 toN altogether. Next, we introduce the concept of connectedness in a poset. This concept will be discussed later in much greater detail. We say a poset (X,::;) is connected if its Hasse diagram is connected as a graph.

From the Hasse diagram of the poset we can count the ways, we say path, from one point to another point of length i, where length means the number of line segments in the path between two points in the Hasse diagram of the poset. For instance, the lengths of paths from the point 1 to the point 5 are 2 and 3 respectively in the following Hasse diagram of the poset. 8) Using the adjacency matrix we can count the number of paths from any given point to enery other point in the poset more efficiently.