In reference to airspace, what does the FAA mean by "within the lateral boundaries of"? Is this construed to also imply a vertical limit?
The FAA normally interprets the phrase "within the lateral boundaries of" (x airspace) to mean any airspace whose lat-lon coordinates fall within said boundaries, regardless of altitude, from the earth's surface all the way to outer space, or at least to the upper limits of the airspace which the FAA has any regulatory authority over.

Pilots would be wise to follow this common-sense understanding of the meaning of "within the lateral boundaries of".

However, the FAA is not a monolithic entity, nor is it immune to making the occasional blunder.  On a few occasions, some subset of the FAA has adopted a very different understanding of the meaning of "within the lateral boundaries of".  

For example, in the text of a spectacularly misguided Final Rule published on page 51966 of the October 5, 1993 edition of the Federal Register, the FAA implied that the phrase "within the lateral boundaries of the surface areas of ... Class E airspace designated for an airport" would not encompass any Class E airspace higher than the floor of the 700' or 1200' AGL "transition area" that invariably overlies a Class-E-to-surface airspace.  

To correct this imaginary problem, the FAA changed the wording of FAR 91.157 from the existing wording 


  91.157 Special VFR weather minimums Except as provided in appendix D, section 3 of this part, the following special weather minimums and
  requirements apply to operations conducted to or from an airport in
  controlled airspace:  (a) Operations may be conducted only under an
  ATC clearance-- (1) Within the lateral boundaries of the surface areas
  of Class B, Class C, Class D, or Class E airspace designated for an
  airport; and ...


To the new wording


  91.157  Special VFR weather minimums ( a ) Except as provided in appendix D,
  section 3, of this part, special VFR operations may be conducted under
  the weather minimums and requirements of this section, instead of
  those contained in 91.155, below 10,000' feet MSL within the airspace
  contained by the upward extension of the lateral boundaries of the
  controlled airspace designated to the surface for an airport. (b) 
  Special VFR operations may only be conducted--(1) With an ATC
  clearance...


However, the FAA failed to also change the similar wording of FARs 45.22(a)(3)(ii), 91.155(d), 91.303(c), 91.309(a)(4), 93.152, 101.33(a), 103.17, 107.41, 135.205(b), and 137.43(a), thus implying that none of these FARs should be understood to apply to any Class E airspace above 700' AGL in most cases, or above 1200' AGL in the remaining cases.

This blunder has never been officially corrected-- the text published in the October 5, 1993 edition of the Federal Register has never been officially retracted.