**Order Groups**

** **

Sets can have symbols in sequence {order group},such as pattern or k-tuple.

The set can have subsets. Subsets are symbolsets in sequences and patterns.

All subsets form order groups. For example,pattern "acg" has subsets NULL, "a", "c","g", "ac", "cg", and "acg". Ordergroups contain null set and pattern. If sets can be circular, the set can havesubsets "ga", "cga", and "gac".

Rules can be that patterns are equivalent overgaps and insertions, so "acg" = "ac gX". Gaps or insertionsize or number can have restrictions.

Two patterns share largest subset. Two patternsshare two largest equivalent subsets at optimum alignment.

To compare patterns, add or remove gaps andinsertions from both patterns to find largest subset. If symbols aredimensions, spaces have maximum number of shared dimensions and minimum numberof new dimensions.

Indexes are part of, and have position in,patterns. Pattern symbols have one or more indices. In pattern"acgta", symbol "a" is at position 1 and 5. Pattern subsetsstart at one or more indices. In pattern "acgta", subset"ac" starts at position 1.

Combining patterns results in new symbolsequences and patterns. Start with first pattern and add new symbols in sequence.Discard symbols that are the same in sequence, for example, "abc" and"ag" nets "abcg". It is like union of sets but with orderin elements.

Combining is associative but not commutative.Null pattern combines with pattern to give same pattern. The same patterncombines with itself to give same pattern. Inverse pattern combines withpattern to give null pattern, but there can be no inverse pattern.

Finding largest aligned subset is like setintersection. Aligning "abc" and "ag" nets "a".

Aligning is associative and commutative. Nullpattern aligns with pattern to give null pattern. The same pattern aligns withitself to give same pattern. Inverse pattern combines with pattern to give nullpattern.

Natural or artificial objects, events, lines,surfaces, solids, n-dimensional figures, geometric points, figures, or imagescan be linear single-symbol series and so can be patterns. Patterns have ordergroups, and so all things can align. For example, letter "a" can standfor angle of 45 degrees and letter "L" can stand for angle of 90degrees, so pattern "aLa" can stand for right triangle with two45-degree angles.

Symbol sequences can transform into symbol groupsequences. For example, pattern "acgta" has three-symbol subsets,"acg", "cgt", and "gta", rather than afive-symbol sequence. Subsets can align rather than single symbols.

Two objects or events can transform into linearRNA-base sequences. They can align by hybridization.

Things can use patterns with few symbols andlong sequences or shorter sequences with more symbols.

Perhaps, brain can compare patterns using ordergroups.

----------------------

Copyright © 2011 John Franklin Moore. All rightsreserved.