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The free product G*H of groups G and H is the set of elements of the form g_1h_1g_2h_2...g_rh_r, where g_i in G and h_i in H, with g_1 and h_r possibly equal to e, the ...

A semigroup with a noncommutative product in which no product can ever be expressed more simply in terms of other elements.

A tree which is not rooted, i.e., a normal tree with no node singled out for special treatment (Skiena 1990, p. 107). Free trees are sometimes known instead as unrooted trees ...

An occurrence of a variable in a logic formula which is not inside the scope of a quantifier.

Two closed simply connected 4-manifolds are homeomorphic iff they have the same bilinear form beta and the same Kirby-Siebenmann invariant kappa. Any beta can be realized by ...

A group acts freely if there are no group fixed points. A point which is fixed by every group element would not be free to move.

The Freemish crate, also called Escher's cube (Elber) or Hyzer's illusion (Pappas 1989, p. 13), is an impossible figure box that can be drawn but not built. It appears in ...

A strophoid of a circle with the pole O at the center of the circle and the fixed point P on the circumference of the circle. Freeth (1878, pp. 130 and 228) described this ...

Pick any point P on a conic section, and draw a series of right angles having this point as their vertices. Then the line segments connecting the rays of the right angles ...

The end of the last gap in the Lagrange spectrum, given by F=(2221564096+283748sqrt(462))/(491993569)=4.5278295661... (OEIS A118472). Real numbers greater than F are members ...