By Frederick S. Woods

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**Example text**

The closed sets are the sets of the form P for the first, and O for the second operator. Þ Þ Þ Þ ß Þ Þ Þ✙à Þ Þ Algebras and Structures Þ 15 ß Þ ß O . ßOnß the other hand, O ⑥ ✱ ß ß which O Proof. We have O ⑥ O , from ✱ O , so that we get O O . Likewise, P now follow easily. P is shown. 25 Let ✜ A ✢ Bß ✢ R ✣ be a context. A pair ✜ O ✢ P ✣❀✤ ℘✼ A ✽❨❡ ℘✼ B ✽ is ✱ ✱ called a concept if O P and P O . Þ Þ Þ ß ✜ A ✢ B ✢ R ✣ be a context. 26ß Let of the form ✜ P ✢ P ✣ , P ⑥ B, or, alternatively, the pairs of the form ✜ O ✢ O ✣ , O ⑥ A.

22 Let ✜ A ✢ B ✢ R ✣ be a context. Then the following holds for all O ✢ O → ⑥ A and all PÞ❀✢ àP → ⑥ B. ß Þ à P. ➀ O ⑥ P iff O Þ ➁ If O ⑥ O → then O à O→ . ß ➂ If P ⑥ ÞP → then P ß ß P→ . ➃ O ⑥ OÞ . ß ➄ P⑥ P . Proof. Notice that if ✜ A ✢ B ✢ R ✣ is a context,Þ ✜ B ✢ A ✢ ßR ➑✸✣ also is a context, Þ and so we only need to show ➀, ➁ and ➃. ➀. O ⑥ P iff every x ✤ O stands in relation R to every member of P iff Þ Þ à P ⑥ Þ O . ➁. If O ⑥ O → and Þ y ✤ O → , then ß which is the for every x ✤ O → : x R y.

We give another example. The combination of the letters ❋✵✴ is pronounced in two noticeably distinct ways in German. After [ı], it ✱ ✧ sounds like [c¸], for example in ✶ ❋✵✴ [lıc¸t], but after [a] it sounds like [x] as ✵ ✧ in ✷ ❋★✴ [naxt]; the choice between these two variants is conditioned solely by the preceding vowel. It is therefore assumed that German does not possess 32 Fundamental Structures Table 1. German Plural Morphs singular ✵ ☛❉✞✬ ✪✘✧ ❈ ✪✒✮ ✸ ✒✹ ✺ ✻ ✱ ✧ ✼ ✵✗✧✘✵❋ ✴ ❉✫★ ✷ ❋✵✴ ✶ ✵ ✧ plural ✵ ✘❉✹✬ ✺ ✪✍✧ ❈ ✮ ✻ ✪✳✮☛✮✹❉ ✱ ✧✎❉✫★ ✶ ❋✬✵ ✧✎✴ ❉❃★ ★ ✼✾✽ ✵ ✽ ✧✎❉ ✷ ❋ ✴ ★ ✸ ✗✹ ‘car’ ‘car’ ‘bus’ ‘light’ ‘father’ ‘night’ two phonemes but only one, written ❋✵✴ , which is pronounced in these two ways depending on the context.

### A course in mathematics,: For students of engineering and applied science, by Frederick S. Woods

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