The notion of a pre-period of an algebra A is defined by means of the notion of the pre-period 位(f ) of a monounary algebra A; f : it is determined by sup{位(f ) | f is an endomorphism of A}. In this paper we focus on the pre-period of a finite modular lattice. The main result is that the pre-period of any finite modular lattice is less than or equal to the length of the lattice; also, necessary and sufficient conditions under which the pre-period of the glued sum is equal to the length of the lattice, are shown. Moreover, we show the triangle inequality of the pre-period of the glued sum.
The pre-period of a finite algebra is the maximum pre-period of its endomorphisms. We know that the pre-period of any finite modular lattice is less than or equal to the length of the lattice. A finite modular lattice is said to have the maximum pre-period property (MPP) if its pre-period and its length are equal. In this paper, we study MPP of the direct product of chains.
The notion of a pre-preriod of a finite bounded distributive lattice (BDL) A is defined by means of the notion of a pre-period of a finite connected monounary algebra: it is the maximum value of the pre-period of an endomorphism and 0-fixing connected mapping of A to A. The main result is that the pre-period of any finite BDL is less than or equal to the length of the lattice; also, necessary and sufficient conditions under which it is equal to the length of the lattice, are shown.
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